From solovay@math.berkeley.edu Sat Apr 15 16:00 MDT 1995 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA03546; Sat, 15 Apr 1995 15:59:59 -0600 Return-Path: Received: by math.berkeley.edu (8.6.10/1.33(math)Ow.1) id OAA23185; Sat, 15 Apr 1995 14:55:33 -0700 Date: Sat, 15 Apr 1995 14:55:33 -0700 From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199504152155.OAA23185@math.berkeley.edu> To: holmes@catseye.idbsu.edu Subject: First of a series of letters Status: RO Randall, It is raining here in Oakland, and so I write on my Powerbook [whose connection with the internet is somewhat flaky.] Partly as a consequence of this flakiness, I will be composing a series of short letters rather than one jumbo letter. The first installment concerns the following question which arose during my musings recently about proving the consistency of NFU. Of course, thanks to Specker, we know that NF refutes the axiom of choice. But does it rule out the assertion that every well-founded set [one arising from a well-founded extensional relation] is constructible? I couldn't see that it does. More to come. --Bob From solovay@math.berkeley.edu Sat Apr 15 17:21 MDT 1995 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA03577; Sat, 15 Apr 1995 17:21:13 -0600 Return-Path: Received: by math.berkeley.edu (8.6.10/1.33(math)Ow.1) id QAA24218; Sat, 15 Apr 1995 16:16:47 -0700 Date: Sat, 15 Apr 1995 16:16:47 -0700 From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199504152316.QAA24218@math.berkeley.edu> To: holmes@catseye.idbsu.edu Subject: An example Status: RO I mentioned in a previous letter that I had an example of a set of sentences Sigma which satisfied your suggested conditions but such that the resulting term model did not satisfy the axiom of counting. My original example employed both an inaccessible cardinal and the fine structure technology of Jensen. I've since realized that the example can be simplified so as not to use the Jensen technology. The question under consideration is an arithmetic one, so we may wlog assume that V=L. {Though I'm not really using this assumption.] Let alpha be an inaccessible cardinal. We are going to choose an ultrafilter U on alpha. Using this U and its powers and the structure L_alpha will generate a set of sentences Sigma. We have to insure that there will be a definable function f from alpha to omega which is non-constant mod U. This will, as in the previous example, insure that the term model generated by U and the definable functions of the structure L_alpha will not satisfy counting. The function f may be described as follows. Let beta be an ordinal. Then beta can be written as a sum lambda + n, where lambda is 0 or a limit ordinal, and n is a non-negative integer. Then f(beta) will be this n. It is routine to construct a uniform ultrafilter U on alpha such that f is non-constant mod U. This completes the description of the revised construction. As ever, Bob From solovay@math.berkeley.edu Sat Apr 15 18:02 MDT 1995 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA03606; Sat, 15 Apr 1995 18:02:48 -0600 Return-Path: Received: by math.berkeley.edu (8.6.10/1.33(math)Ow.1) id QAA24836; Sat, 15 Apr 1995 16:58:22 -0700 Date: Sat, 15 Apr 1995 16:58:22 -0700 From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199504152358.QAA24836@math.berkeley.edu> To: holmes@catseye.idbsu.edu Subject: Some large cardinal properties Status: RO Randall, This letter just defines some concepts that I need to state the results in my following letter on upper bounds on the consistency strength of some variants of NFU. Here are some large cardinal properties arranged in order of strictly decreasing consistency strength. The ones that occur after 0# are all compatible with V=L. 1) strongly compact 2) measurable 3) 0# exists 4) Erdos 5) completely ineffable 6) ineffable 7) weakly compact 8) Mahlo 9) inaccessible The ones that I need for the next letter are 4) and 5). I should caution that I am not sure that I am using the term "completely ineffable" as it is used in the literature. In the letter that follows I am using it in the sense to be given below. The definition of an Erdos cardinal is somewhat delicate. Instead I describe a property such that the least cardinal with that property is the least Erdos cardinal. Here is the property in question. Whenever A is a structure [in a countable similarity type] with underlying set the cardinal kappa, then there is a set of indiscernibles for the structure A of order type omega. The definition of "completely ineffable" is in terms of a certain two person game. In this game, player II plays subsets of kappa, A_0, A_1, ... These are to be a decreasing sequence of subsets each of cardinality kappa. If ever player II cannot play a legal move, then he loses; if he never fails to play a legal move, he wins the game. To start things off, A_{-1} is the set of limit ordinals less than kappa. Player I has two sorts of moves he can play. 1) He can play an array of length kappa of subsets of kappa. To respond II plays a set which is [modulo sets of cardinality less than kappa] either contained in or disjoint from each set of the array. 2) I can play a regressive function f on A_{n-1}. That is, f(beta) < beta for any beta in A_{n-1}. II must play a set A_n on which f is constant. This completes the description of the game. If II wins, kappa is completely ineffable. As an instructive exercise show that if kappa is measurable, then kappa is completely ineffable. As ever, Bob From solovay@math.berkeley.edu Sat Apr 15 18:11 MDT 1995 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA03620; Sat, 15 Apr 1995 18:11:53 -0600 Return-Path: Received: by math.berkeley.edu (8.6.10/1.33(math)Ow.1) id RAA24929; Sat, 15 Apr 1995 17:07:26 -0700 Date: Sat, 15 Apr 1995 17:07:26 -0700 From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199504160007.RAA24929@math.berkeley.edu> To: holmes@catseye.idbsu.edu Subject: Some upper bounds on consistency strength Status: RO We will be considering two extensions of NFU: NFUA: NFU + Infinity + Choice plus Every Cantorian set is strongly cantorian. NFUB: NFUA + the axiom that seys every definable subset of the strongly cantorian well-founded sets is the intersection of the well-founded strongly cantorian sets with some set of the model. Result 1: The consistency of each of these theories is provable in Zermelo set theory plus "There is a completely ineffable cardinal". Result 2: Suppose there is an Erdos cardinal. Then there is an inaccessible cardinal gamma and a model N of NFUB such that the strongly Cantorian sets of this model are isomorphic to V_gamma. [gamma will in fact be completely ineffable and hence weakly compact]. It is interesting to note that I can't get a model of NFUA from any assumption that doesn't also yield a model of NFUB. The model of NFUA that I get in Result 1 is countable, but its strongly Cantorian ordinals are well-ordered. As ever, Bob From solovay@math.berkeley.edu Sun Apr 16 17:27 MDT 1995 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA03928; Sun, 16 Apr 1995 17:27:20 -0600 Return-Path: Received: by math.berkeley.edu (8.6.10/1.33(math)Ow.1) id QAA07568; Sun, 16 Apr 1995 16:22:53 -0700 Date: Sun, 16 Apr 1995 16:22:53 -0700 From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199504162322.QAA07568@math.berkeley.edu> To: holmes@catseye.idbsu.edu Subject: Lower bounds Status: RO Here is a theorem I am willing to claim: NFUA proves the consistency of "ZFC + There is an inaccessible limit of inaccessibles". It's not quite clear what the best I can get is. I definitely haven't yet gotten the consistency of "ZFC + There is a Mahlo cardinal". But I'm still trying. The results claimed contradict, of course, your claim that NFUA is equiconsistent with ZFC [unless ZFC is inconsistent]. I will continue to think about this but this is the last letter I will post until I get some reaction from you. --Bob From solovay@math.berkeley.edu Wed Apr 12 13:48 MDT 1995 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA03920; Wed, 12 Apr 1995 13:48:05 -0600 Return-Path: Received: by math.berkeley.edu (8.6.10/1.33(math)Ow.1) id MAA04011; Wed, 12 Apr 1995 12:44:03 -0700 Date: Wed, 12 Apr 1995 12:44:03 -0700 From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199504121944.MAA04011@math.berkeley.edu> To: holmes@catseye.idbsu.edu Cc: T.Forster@pmms.cam.ac.uk, holmes@catseye.idbsu.edu In-Reply-To: Randall Holmes's message of Wed, 12 Apr 1995 03:41:37 -0600 <199504120938.CAA21877@math.berkeley.edu> Subject: From Randall Holmes Status: RO Dear Randall, 1. You write: The requirement for a model of the axiom of counting is not that each ordinal be standard (sorry, I mean each natural number there) but that each natural number be fixed by the automorphism. Reply: I am **well** aware of this; I explicitly indicated a natural number moved by the automorphism. I will discuss this point again later in this letter. 2. I can now prove a bit more than I asserted in my previous letter. I asserted there that it seemed unlikely that the conditons you imposed on Sigma sufficed to get a model of the Counting axiom [much less the Cantorian axiom]. I can now [assuming an inaccessible cardinal] produce a specific example of a term-model constructed from a Sigma meeting your requirements that does not satisfy the axiom of counting. I will not include a proof of this latter result in this letter, since it involves ideas from Jensen's work on the fine structure of L with which you may not bew familiar. 3. You say that you are fairly certain that the model below an inaccessible with ultrafilters is correct. I **know for certain** that it is not. I will try to convey the proof once again in the hope that this time I will supply sufficient detail to convince you. Please read what I am writing. I am getting the feeling that I am wasting my time in writing to you since you don't read carefully and **think** about what I send. 4. So lets start the discussion of the model "below an inaccessible". alpha is a strongly inaccessible cardinal fixed once for all. We fix an ultrafilter U on alpha, that gives each bounded subset of alpha measure zero. [I am taking the von Neumann approach where each cardinal is an ordinal and each ordinal is the set of all smaller ordinals.] 5. I will sketch an alternate construction of the ultrafilters U_n [for n in omega]. It really is completely equivalent to what you do [though I shall not stop to prove this]. But the approach I will follow is much easier to compute with. So I need first the notion of the cartesian product of two ultrafilters. Suppose that F and G are ultrafilters on sets X and Y respectively. I am going to define an ultrafilter F \cross G which will live on the cartesian product set X \cross Y. Let then A be a subset of X \cross Y. Let f be the characteristic function of A. If we integrate f with respect to its second variable [using G], we get a 2-valued function on X, say f'. If we integrate f' [using F] we get a number in {0,1}. Put A in F \cross G iff the number is 1. This is a standard construction in the theory of ultrafilters. I note only that the order of integration is important. 3. We can now define the U_n's. U_1 is the isomorphic copy of U obtained using the obvious isomorphism of alpha with alpha^1. U_{n+1} is the isomorphic copy of U_n \cross U_1 using the obvious isomprphism [given by concatentation] of alpha^n \cross alpha with alpha^{n+1}. It is easy to see that these U_n's have the various desired properties. In particular, they satisfy the sentences imposed in Sigma. 4. We can form the limit ultraproduct construction [using all the functions from alpha^Z to V_alpha of finite support] as outlined in your paper. To show that the axiom of counting fails in the model, it suffices to find an integer of the model moved by the automorphism j. But this is easy to do. Let F be a map from alpha to omega that is non-constant mod U. Let F_0 be the map of alpha^Z to omega which sends a sequence s to F(s(0)). Similarly, let F_1 be the analgous map that sends the sequence s to F(s(1)). Then the automorphism j sends the equivalence class of F_0 to the equivalence class of F_1. It only remains to see that these equivalence classes are unequal. This amounts to seeing that the U_2 measure of the set of all pairs with F(gamma_0) <> F(gamma_1) is 1. But this is immediate from the way that U_2 was defined and the fact that F is unequal [mod U] to a constant function. As ever, Bob Solovay who has not forgotten **everything** that he learned in his former incarnation as logician. From solovay@math.berkeley.edu Mon Apr 17 12:20 MDT 1995 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA04726; Mon, 17 Apr 1995 12:20:14 -0600 Return-Path: Received: by math.berkeley.edu (8.6.10/1.33(math)Ow.1) id LAA23741; Mon, 17 Apr 1995 11:15:45 -0700 Date: Mon, 17 Apr 1995 11:15:45 -0700 From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199504171815.LAA23741@math.berkeley.edu> To: holmes@catseye.idbsu.edu Cc: holmes@catseye.idbsu.edu In-Reply-To: Randall Holmes's message of Mon, 17 Apr 1995 11:26:22 -0600 <199504171721.KAA21538@math.berkeley.edu> Subject: Lower bounds Status: RO Dear Randall, You write: I will say that it looks to me as if your claim that my model of NFUA using an inaccessible doesn't work springs from a misunderstanding of the way in which the model is constructed. What I would like to do (and will start doing unless you object) is to go through the construction of a model of NFUA given an inaccessible in detail in Reply: This sounds like a fine way to proceed. I don't have a reference for "there is a completely ineffable cardinal below a measurable, though I think its well-known. I proved it by reconstructing the proofs of the following results: Theorem(Silver?) There is an Erdos cardinal less than the first measurable. Theorem(Silver) The least Erdos cardinal, if it exists, is Erdos in L. Theorem(???) There are a stationary set of completely ineffable cardinals below the first Erdos cardinal. I know a reference for the second theorem [as well as its proof.] The first theorem is really easy. As for the third, I cooked up a proof, but I suspect that it's in whatever paper the concept is first introduced in. [I don't recall when or where this paper appeared or by whom the paper was done.] As ever, Bob From holmes Mon Apr 17 12:26:50 1995 To: solovay@math.berkeley.edu Subject: Various Status: RO Dear Bob: I must apologize! You are absolutely right in your objections to the Refinement of Construction 2 as a way to construct models of NFU with even the Axiom of Counting! It isn't that I haven't been reading what _you_ wrote carefully; I haven't been reading what _I_ wrote myself carefully! For I do not claim in the paper that the Refinement of Construction 2 gives models of the Axiom of Counting; all I claim is that (in the case \beth{\alpha} = \alpha, which I will assume from here on out) any ordinal greater than all standard ordinals is moved upward by the automorphism (in the model of NFU resulting, T{\alpha} < \alpha unless \alpha is bounded by a standard ordinal). This is a technically very useful but not very strong property. I knew perfectly well when I was writing this that this did not give models of Counting; to see how I get models of Counting (or \beta strongly cantorian for any fixed \beta < \alpha) see Construction 3 (which I will summarize in a following note, jut to get it completely clear in my head!). The construction of a model of NFUA using an inaccessible (which I still think is possible) is not found in this paper. I'm really sorry about this; I definitely must have appeared (and was being) pig-headed, because I was not remembering what I had done correctly! More stuff follows (an account of Construction 3) but I want to send this off now since we seem to be talking in real time and I want to clear the air of smoke a bit and get down to stating things correctly! I shouldn't have tried to discuss any of this while attending the conferences in Europe, I think; my mind was largely elsewhere :-( --Randall From holmes Mon Apr 17 14:11:48 1995 To: solovay@math.berkeley.edu Subject: Construction 3 Status: RO Dear Bob, Now, hopefully, you are talking to the real Randall Holmes with brain completely engaged :-) Topic 1: Modelling counting (not yet NFUA!) I am going to describe the basic construction of a model of NFU with a certain standard infinite cardinal \beta (you may think of it as \omega, so that this will be a model of Counting, but I will be general, since generality is cheap here) strongly cantorian. It is sufficient to build a model with automorphism in which each ordinal less than \beta (and \beta itself) is fixed by the automorphism. Choose a cardinal \alpha greater than each iterated exponential of \beta (so that the Erdos-Rado theorem can be used). Functions from \alpha to \beta can be coded by elements of \alpha; clearly, so can finite sets of such functions. Consider, for each function f from \alpha to \beta, the set C_f of all codes of finite sets of maps from \alpha to \beta which contain f. The collection of C_f's is a filter on \alpha which can be extended to an ultrafilter C. C "describes" the code of a "finite set" of maps \alpha -> \beta which contains all standard such maps, in the ultrafilter model constructed using C. This "finite set" of functions induces a partition of \alpha into \beta pieces, which has a "homogeneous set" H. We use this "homogeneous set" to define a sequence of ultrafilters U_i on [\alpha]^i in the real world: U_i will contain all standard finite sets of subsets of \alpha of size i whose analogues in the model built with C contain all subsets of H with i elements. The model can be defined in terms of the U_i's as in the earlier constructions. The model contains \beta_k's indexed by integers k: U_i tells us which sets contain each sequence of i successive \beta_k's; the model elements are exactly the images under standard functions of finite sequences of successive \beta_k's. Any ordinal less than \beta in this model is the image of a finite sequence of n successive \beta_k's under some standard function \alpha^n -> \beta. Now the set H in the model built using C was homogeneous with respect to the partition induced by this map (and every similar standard map), so the images of any sequence of n successive \beta_k's remains the same when the indexes of the \beta_k's are incremented; any ordinal less than \beta is fixed by the automorphism. If you have trouble seeing that this information is coded in the U_i's, consider that the information that f(beta_0...beta_{n-1}) = f(beta_1...beta_n) for any standard f and n is coded in U_{n+1}. (I assume that you will _not_ have trouble seeing this; I include it mostly to remind _myself_). This construction gives models of Counting and higher axioms of the form "Ordinal so-and-so is strongly Cantorian". It does not, of course, give us a basis for any assertion about _all_ Cantorian or s.c. sets. I seem not to have included an argument for a model of NFUA using an inaccessible in the paper as it stands; I'm going to have to reconstruct this! It was a refinement of this argument for models with s.c. sets, and I seem to have thought that it was redundant given the sharper result for ZFC which I have not communicated successfully! I will look at old drafts of the paper and see if I can find that construction; meanwhile, do you believe this construction? Topic 2: A possible description of the set of sentences needed to establish that Con(ZFC) = Con(NFUA) As in the argument in the paper, we work in a term model of ZFC. Add alpha_i's, nonstandard ordinals, indexed by integers (levels can be recovered as V_{\alpha_i's}) They satisfy the following sentences: G(\alpha_1...\alpha_n) < \alpha_{n+1}, G standard For each partition P of finite subsets of the ordinals of size n into set many compartments, definable using a finite sequence of \alpha_i's whose largest index is k, all \alpha_i's with index higher than k belong to the same cofinal homogeneous class for P. The question about the latter collection of sentences (in my mind) is again whether it is expressible without essential reference to classes. I'll take another stab at expressing it: Let F(\alpha_{k-m}...\alpha_k) be a class map from n-element subsets (represented as ascending sequences) of the ordinals onto some (von Neumann) ordinal (i.e., it has bounded values, and so is a partition of the kind indicated above). The part of the definition of this partition represented by the letter F is standard (in the term model). Then we have F(\alpha_{j_1}...\alpha_{j_n}) = F(\alpha_{k_1}...\alpha_{k_n}) whenever all j_m's, k_m's > k. I haven't directly provided that the homogeneous class is cofinal (which is essential for iteration; one doesn't want it to become small!), but this seems to be provided by the first set of sentences; any class which contains all the \alpha_i's seems to have to be cofinal. But this condition seems to be expressible as well (without essential reference to classes): to say that a given homogeneous class can be regarded as cofinal is to say that the corresponding value for F is found on homogeneous sets relative to this partition of arbitrarily large ordinals, I think? Do you agree that this set of sentences is describable and consistent (I do say something like this in the paper)? Existence of cofinal homogeneous classes does follow from Erdos-Rado, does it not? It appears that this set of sentences is sufficient for my purposes (the "standard-bounded" stuff is not needed). Compare any \alpha_i with a standard function image F(\alpha_{i+1}...\alpha_{i+n}). Consider the partition produced by mapping each sequence (a_1...a_n) to F(a_1...a_n) if this is less than \alpha_i and to \alpha_i otherwise. The larger \alpha_k's must be homogeneous with respect to this partition: thus, F(\alpha_{i+1}...\alpha_{i+n}) is either (i.) greater than or equal to \alpha_i or (ii.) lies below \alpha_i and is fixed under the automorphism induced by incrementing the indices of the \alpha_i's (that this operation is an automorphism follows from the homogeneity properties of the model). Thus, in the induced model of NFU (built from V_{\alpha_0) using the index incrementing automorphism), every ordinal will either be fixed under the automorphism or greater than some \alpha_i; this is sufficient for the Axiom of Cantorian Sets to hold. Final Remark: Now that I have what I think is the correct set of sentences for the Con(ZFC) -> Con(NFUA) proof, I think I see how to reconstruct the model of NFUA below an inaccessible. But I will get this part out before tackling the latter in the next message! I hope that I am being more coherent! --Randall From holmes Mon Apr 17 14:48:07 1995 To: holmes@catseye.idbsu.edu, solovay@math.berkeley.edu Subject: Re: Construction 3 Status: RO I'm not sure that I stated exactly how the "finite set" of partitions described by the ultrafilter C in the outline of Construction 3 is used to induce a single partition for which a homogeneous set can be found. If this is not clear, ask me. --Randall From solovay@math.berkeley.edu Tue Apr 18 23:39 MDT 1995 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA07457; Tue, 18 Apr 1995 23:39:52 -0600 Return-Path: Received: by math.berkeley.edu (8.6.10/1.33(math)Ow.1) id WAA09173; Tue, 18 Apr 1995 22:35:23 -0700 Date: Tue, 18 Apr 1995 22:35:23 -0700 From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199504190535.WAA09173@math.berkeley.edu> To: holmes@catseye.idbsu.edu Subject: Upper bounds on NFUB's consistency strength Status: RO For a certain new property Phi(kappa), I can prove ZFC + exists kappa Phi(kappa) proves Con(NFUB). Before giving the precise definition let me make some comments. 1) If kappa is completely ineffable, then there are a stationary set of alpha's less than kappa such that Phi(alpha). 2) It follows that there are alpha in L such that Phi(alpha) [say if 0# exists]. But I cannot prove yet that if Phi(alpha), then Phi(alpha) holds in L. Indeed, for all I know, "ZFC + V=L + exists alpha Phi(alpha)" has greater consistency strength than "ZFC + exists alpha Phi(alpha)". [I don't think this actually happens; I just can't yet rule it out.] 3) If Phi(alpha), then alpha is weakly compact, and there are a stationary set of beta less than alpha which are n-ineffable for all n in omega [and hence are weakly compact]. 4) The idea of Phi(alpha) is to give just as much of the normal measure coming from a measurable as is needed in [my version of] the usual proof that measurable cardinals give the consistency of NFUB. Here is the precise definition: A cardinal kappa is Phi provided there is a well-ordering of V_kappa of length kappa, <*, say, a non-empty collection of subsets of kappa, say W, and a finitely additive ultrafilter U on W such that: (a) To avoid any confusion I spell out what I mean by an ultrafilter U on W. It will follow from the closure conditions on W asserted below that W is a Boolean subalgebra of P(kappa). There is a Boolean homomorphism of W onto 2 such that U is the preimage of 1. Note well: we do not assume that U is countably additive. (b) Here are the explicit closure conditions on W. W contains every one element subset of kappa. If A and B are members of W and C is a subset of kappa which is definable in the structure then C is in W. (c) Suppose that is a kappa sequence of subsets of kappa that is coded by a member of W> Then the set of alpha such that A_alpha is in U is a member of W. [This says that U, in the terminology of Kunen, is a W-ultrafilter.] Finally, we need the following normality condition. (d) Suppose that A is in U, and that f is map from A to kappa, which is regressive in the sense that f(alpha) < alpha for every alpha in A. Then there is a subset B of A, also lying in U, such that the restriction of f to B is constant, That completes the definition of Phi and with it this letter. As ever, Bob From solovay@math.berkeley.edu Wed Apr 19 10:36 MDT 1995 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA07895; Wed, 19 Apr 1995 10:36:36 -0600 Return-Path: Received: by math.berkeley.edu (8.6.10/1.33(math)Ow.1) id JAA17901; Wed, 19 Apr 1995 09:32:05 -0700 Date: Wed, 19 Apr 1995 09:32:05 -0700 From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199504191632.JAA17901@math.berkeley.edu> To: holmes@catseye.idbsu.edu In-Reply-To: Randall Holmes's message of Wed, 19 Apr 1995 06:33:16 -0600 <199504191228.FAA13457@math.berkeley.edu> Subject: Net effect Status: RO Randall, This letter is being written in great haste early in the morning in response to your two most recent letters. 1) It is possible that my "quite erroneous" appelation of your first proof was "quite erroneous". I will have to think about this. What I am pretty sure is that your ultrafilter style arguments do not yield the Jensen-Morley result that for any standard beta, NFU has a beta model. When I read this proof, I thought this was what you were trying to prove. So I will have to read the proof again to see if it yields a proof of the consistency of NFU + Counting. 2) I will be glad to (a) discuss my objections to your proof [if they survive a second reading thereof] and your "proof" of Con(ZFC) implies Con(NFUA). But the next thing I want to tackle [in our correspondence] is the typing in of my proof that NFUA implies the consistency of ZFC {and a fair bit more!] I have no time right now to read and comment on the mathematics in your two latest letters. All in due time. As ever, BobB From solovay@math.berkeley.edu Wed Apr 19 13:38 MDT 1995 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA08078; Wed, 19 Apr 1995 13:38:28 -0600 Return-Path: Received: from feynman.berkeley.edu by math.berkeley.edu (8.6.10/1.33(math)Ow.1) id MAA23808; Wed, 19 Apr 1995 12:33:58 -0700 From: solovay@math.berkeley.edu (Robert M. Solovay) Received: (solovay@localhost) by feynman.berkeley.edu (8.6.10/8.6.4) id MAA12316; Wed, 19 Apr 1995 12:33:57 -0700 Date: Wed, 19 Apr 1995 12:33:57 -0700 Message-Id: <199504191933.MAA12316@feynman.berkeley.edu> To: holmes@catseye.idbsu.edu Subject: On the consistency strength of NFU + Counting Status: RO Of course, ZFC simply proves NFU + Counting consistent. So we have to use weaker theories to calibrate its consistency strength. The following terminology is pretty standard: Z is Zermelo set theory. [So we drop the replacement axiom and the axiom of choice; we do have a comprehension schema. ZC is Zermelo plus the axiom of choice. ZF- is ZFC - the power set and choice axioms. ZFC- is ZFC minus the power set axiom. The weakest theory in which I can prove the consistency of NFU + "Counting" is ZFC- + "Aleph_{Aleph_{Aleph_1}} exists". If I weren't so lazy, I could almost certainly hack out a proof in ZC + "For all countable ordinals alpha, Aleph_{Aleph_alpha} exists". How about lower bounds. Well we certainly can get the consistency of ZC + V=L + "Aleph_alpha exists" for any explicit small countable ordinal alpha. [For example, alpha = omega^2.] I think with a little work, this could be greatly improved---to for example, ZC + V=L + Aleph_{Aleph_{omega^2}. The idea would be to exploit the examples showing the cardinal bounds for Erdos-Rado are optimal. As I indicate, I haven't thought this through. What is the relevance of all this. Well so far as I can see, your proof if it were correct could easily be formalized in ZC + V=L + Aleph_omega exists. [Your beta is omega; your alpha could be taken to be Aleph_omega.] In this way, we would easily contradict Godel's theorem. As ever, Bob From solovay@math.berkeley.edu Wed Apr 19 14:16 MDT 1995 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA08116; Wed, 19 Apr 1995 14:16:18 -0600 Return-Path: Received: from feynman.berkeley.edu by math.berkeley.edu (8.6.10/1.33(math)Ow.1) id NAA25005; Wed, 19 Apr 1995 13:11:48 -0700 From: solovay@math.berkeley.edu (Robert M. Solovay) Received: (solovay@localhost) by feynman.berkeley.edu (8.6.10/8.6.4) id NAA12388; Wed, 19 Apr 1995 13:11:46 -0700 Date: Wed, 19 Apr 1995 13:11:46 -0700 Message-Id: <199504192011.NAA12388@feynman.berkeley.edu> To: holmes@catseye.idbsu.edu Subject: Lower bound for strength of NFUA: 1 Status: RO Just some organizational comments and caveats. I envision a series of five letters of which this is the first. Letter 2 will just state precisely what I am claiming. It is a tiny bit stronger than the off-hand claims I made in previous letters. The proof requires a fair amount of reasoning within NFUA. I prefer to present this as follows. I reformulate NFUA as a Specker theory and work within that. The advantage, for me, is that my usual set-theoretical intuitions are readily available. One could, no doubt, do the same argument within orthdox NFUA. Letter 3 gives the main construction, which involves defining two descending sequences of ordinals. The definition takes place internally to the Specker model, so at some point the construction must break down. Letter 4 gives a proof that the construction never breaks down. Although it now seems quite straightforward to me, this is the trickiest part of the argument. I suspect that if my proof is wrong [and I don't think it is] then here is where the error lies. Of course, Letters 3 and 4 together yield a contradiction but to what? Well before starting the argument in Letter 3 an "anti-large-cardinal" assumption was made, and that is what's refuted. A final caveat: This is the first time I've written up the proof in detail and I am doing it in real time. There is definitely a risk that when I come to write letter 4, say, I will just say "Whoops". Of course, I could avoid that by writing this all offline, but I choose not to do so. It's psychologically rewarding to send each letter off into the aether, and as a born again lazy person I need all the props and tricks I can muster to get myself to do anything at all. As ever, Bob From holmes@catseye.idbsu.edu Wed Apr 19 14:17 MDT 1995 Received: by catseye.idbsu.edu (1.38.193.4/16.2) id AA08119; Wed, 19 Apr 1995 14:17:04 -0600 Date: Wed, 19 Apr 1995 14:17:04 -0600 From: Randall Holmes Return-Path: To: holmes@catseye.idbsu.edu, solovay@math.berkeley.edu Subject: Re: On the consistency strength of NFU + Counting Status: RO Dear Bob, I have heard the assertion that the consistency strength of NFU + Counting is Z + < aleph-omega replacement; I don't know how to evaluate it. I think that the consistency strength of my construction is higher than you think. It is necessary to construct V_{alpha} for alpha nonstandard just less than aleph-omega (at least, that's where I _think_ the ordinals moved by the automorphism live) to build the model of NFU; so one certainly has aleph(aleph n) for each n, for example! A proof that aleph(aleph(omega)) exists under Counting would Godelize my construction to my satisfaction! One needs to be able to construct the ordinal ranks corresponding to the ordinals moved by the automorphism as well as those ordinals themselves! If you want to talk about this construction, I want to write it up carefully again from first principles; exact descriptions of the various ultrafilters are important! (and my comment that the U_i's are used in a way analogous to those in construction 2 might be misleading; this does not mean that the U_i's have the same properties as those in the earlier construction). The obstruction to the proof for ZFC is that I need a sharper form of the Erdos-Rado theorem, in the sense that the Paris-Harrington theorem is sharper than the Ramsey theorem: I need homogeneous sets which are larger in cardinality than their smallest element. I only realized that I needed something like this this morning, and I'm quite willing to believe that such a theorem would be either definitely false or would strengthen ZFC (by analogy with what the P-H theorem does). --Randall From holmes@catseye.idbsu.edu Wed Apr 19 14:19 MDT 1995 Received: by catseye.idbsu.edu (1.38.193.4/16.2) id AA08129; Wed, 19 Apr 1995 14:19:00 -0600 Date: Wed, 19 Apr 1995 14:19:00 -0600 From: Randall Holmes Return-Path: To: holmes@catseye.idbsu.edu, solovay@math.berkeley.edu Subject: Re: Lower bound for strength of NFUA: 1 Status: RO I'm fine with sending off "proofs in progress"; I'm guilty of this myself! (as witness the "proof" I started this morning; claim 1 foundered on close examination, or at least is nontrivial!) --Randall From solovay@math.berkeley.edu Wed Apr 19 15:41 MDT 1995 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA08285; Wed, 19 Apr 1995 15:41:29 -0600 Return-Path: Received: from feynman.berkeley.edu by math.berkeley.edu (8.6.10/1.33(math)Ow.1) id OAA27834; Wed, 19 Apr 1995 14:36:58 -0700 From: solovay@math.berkeley.edu (Robert M. Solovay) Received: (solovay@localhost) by feynman.berkeley.edu (8.6.10/8.6.4) id OAA12461; Wed, 19 Apr 1995 14:36:57 -0700 Date: Wed, 19 Apr 1995 14:36:57 -0700 Message-Id: <199504192136.OAA12461@feynman.berkeley.edu> To: holmes@catseye.idbsu.edu Subject: Lower bound for strength of NFUA: 2 Status: RO Here I can state the precise result I am heading for fairly briefly. Work in NFUA. Among the well-orderings of V there is a shortest one. Fix a well-ordering of V of that order type. Call that order type Omega. There will be a binary relation on V which is a model of Z-- + V=L and whose ordinals have order type Omega. Here Z-- is either Z or ZF- depending on whether Omega is a limit cardinal in L or not. [This can be talked about since, in effect, we have type-theory with V as the base set.] Main Claim: There is an ordinal of this model which the model thinks is an inaccessible cardinal with an inaccessible limit of inaccessibles below it. >From the fact that NFUA proves the main claim, it follows readily, that NFUA proves the consistency of ZFC + "There is an inaccessible limit of inaccessible cardinals". It remains to prove the main claim in NFUA. We will work in the theory NFUA + "Main Claim is false" and in the letters that follow will derive a contradiction. From solovay@math.berkeley.edu Wed Apr 19 15:29 MDT 1995 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA08267; Wed, 19 Apr 1995 15:29:13 -0600 Return-Path: Received: from feynman.berkeley.edu by math.berkeley.edu (8.6.10/1.33(math)Ow.1) id OAA27409; Wed, 19 Apr 1995 14:24:43 -0700 From: solovay@math.berkeley.edu (Robert M. Solovay) Received: (solovay@localhost) by feynman.berkeley.edu (8.6.10/8.6.4) id OAA12451; Wed, 19 Apr 1995 14:24:42 -0700 Date: Wed, 19 Apr 1995 14:24:42 -0700 Message-Id: <199504192124.OAA12451@feynman.berkeley.edu> To: holmes@catseye.idbsu.edu Cc: holmes@catseye.idbsu.edu In-Reply-To: Randall Holmes's message of Wed, 19 Apr 1995 14:17:04 -0600 <199504192012.NAA25020@math.berkeley.edu> Subject: On the consistency strength of NFU + Counting Status: RO Randall, I suspected you might reply that the issue is Aleph_{Aleph_omega}. I may look at the proposed strengthening of the lower bound; or I may simply confine myself to looking at your proof. One doesn't need Godel, though it can be a useful way of getting insight. To whom is this proof of the consistency strength of Counting attributed? Is it published? I find myself **rather** skeptical. I think it likely that your formulation of P-H variants of Erdos-Rado is somewhat strong. Not too strong, of course; they follow from weakly compact cardinals. But I've never really thought about such matters so my opinion isn't worth too much. As ever, Bob From holmes@catseye.idbsu.edu Wed Apr 19 16:37 MDT 1995 Received: by catseye.idbsu.edu (1.38.193.4/16.2) id AA08321; Wed, 19 Apr 1995 16:37:59 -0600 Date: Wed, 19 Apr 1995 16:37:59 -0600 From: Randall Holmes Return-Path: To: holmes@catseye.idbsu.edu, solovay@math.berkeley.edu Subject: Re: On the consistency strength of NFU + Counting Status: RO It's mine; it appears in an appendix to my Ph.D. thesis. --Randall Do you have references for P_H variants of Erdos-Rado? I don't need full strength! The construction I had in mind to prove Claim 1 works if I have a P-H variant of Erdos-Rado; of course, the strength needed is then not ZFC! --Randall P.S. I will send you the full write-up of Construction 3 that I am doing now shortly. From holmes@catseye.idbsu.edu Wed Apr 19 16:55 MDT 1995 Received: by catseye.idbsu.edu (1.38.193.4/16.2) id AA08351; Wed, 19 Apr 1995 16:55:38 -0600 Date: Wed, 19 Apr 1995 16:55:38 -0600 From: Randall Holmes Return-Path: To: holmes@catseye.idbsu.edu, solovay@math.berkeley.edu Subject: Re: Lower bound for strength of NFUA: 2 Status: RO I believe the assertions in this note, other than the Main Claim which remains to be established, of course! --Randall From solovay@math.berkeley.edu Thu Apr 20 02:15 MDT 1995 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA08687; Thu, 20 Apr 1995 02:15:16 -0600 Return-Path: Received: by math.berkeley.edu (8.6.10/1.33(math)Ow.1) id BAA11727; Thu, 20 Apr 1995 01:10:46 -0700 Date: Thu, 20 Apr 1995 01:10:46 -0700 From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199504200810.BAA11727@math.berkeley.edu> To: holmes@catseye.idbsu.edu Subject: Letter 3 on Lower Bounds Status: RO So we have our model of NFUA where the Main Claim is false. There is a relation eta on V such that V equipped with eta is isomorphic to L_Omega. We now do the Specker trick of unravelling the model. So V_i is the i^{th} copy of V. There is an epsilon relation between V_i and V_{i+1}, say epsilon_i. There is a copy of eta on V_i, which we dub eta_i. V_i equipped with this relation is isomorphic to L_{Omega_i}. There is a natural inclusion map of L_Omega{i} into L_{Omega{i+1}, which is the analogue in this approach of the T construction. This maps L_{Omega_i} onto an initial segment of L_{Omega_{i+1}}. It is a consequence of the Burali-Forti paradox that this inclusion map is not onto. It is important that these inclusion maps prolong to isomorphisms of finite type structures using the epsilon_i's. Moreover, it is legitimate to allow the various inclusion maps and the eta_i's as well as the epsilon_i's [and the notion of which elements of V_{i+1} correspond to subsets of V_i] to appear in the comprehension principles which define [epsilon type subsets of the various V_i's. There is of course an automorphism j which sends eta_i onto eta_{i+1} for any i, and sends V_i onto V_{i+1} and also carries epsilon_i} onto epsilon_{i+1}. It also maps canonical inclusion maps to one another in the obvious way. We treat the various inclusion maps as identifications for the most part. With these conventions, the ordinals Omega_j for j < i appear in V_i. Moreover, j(Omega_i) = Omega_{i+1}. All this is fairly straightforward. The only three caveats: 1) The comprehension axioms involve only variables ranging over the V-i's one at a time; there are no global variables ranging over the directed limit of the L_{Omega_i}'s. 2) There may well be non-constructible subsets of sets in the various L_{Omega_i}'s . 3) j cannot appear in the instances of the comprehension axiom. I trust all this is utterly clear to you. I will abuse notation by referring to all the different epsilon_i's and eta_i's as epsilon. This just eases the writing; it would be easy to get all the subscripts right if it were important. This ends letter 3. From holmes@catseye.idbsu.edu Thu Apr 20 07:32 MDT 1995 Received: by catseye.idbsu.edu (1.38.193.4/16.2) id AA08739; Thu, 20 Apr 1995 07:32:27 -0600 Date: Thu, 20 Apr 1995 07:32:27 -0600 From: Randall Holmes Return-Path: To: holmes@catseye.idbsu.edu, solovay@math.berkeley.edu Subject: Re: Letter 3 on Lower Bounds Status: RO All of this is clear to me as you expected. --Randall From T.Forster@pmms.cam.ac.uk Thu Apr 20 12:18 MDT 1995 Received: from emu.pmms.cam.ac.uk by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA09004; Thu, 20 Apr 1995 12:18:16 -0600 Return-Path: Received: by emu.pmms.cam.ac.uk (UK-Smail 3.1.25.1/1); Thu, 20 Apr 95 18:58 BST Message-Id: Date: Thu, 20 Apr 95 19:13 BST From: Thomas Forster To: holmes@catseye.idbsu.edu Subject: Re: Not fixed Status: RO I can't think of any reason why ``every cantorian is stcan" should give you anything like zf. From holmes@catseye.idbsu.edu Thu Apr 20 13:54 MDT 1995 Received: by catseye.idbsu.edu (1.38.193.4/16.2) id AA09119; Thu, 20 Apr 1995 13:54:03 -0600 Date: Thu, 20 Apr 1995 13:54:03 -0600 From: Randall Holmes Return-Path: To: T.Forster@pmms.cam.ac.uk, holmes@catseye.idbsu.edu Subject: Re: Not fixed Status: RO Dear Thomas, It seems established that it gives the strength of ZFC as a lower bound; I think that part of my paper is sound. What Solovay claims and I cannot see how to exclude is that it is in fact considerably stronger. The proposition he is now attempting to prove to me (we are not far enough along for me to express an opinion as to the validity of the argument) is that NFU + Infinity + AC + "every Cantorian set is strongly Cantorian" proves that there is an inaccessible limit of inaccessibles in L (as encoded suitably in Ord, not L in the usual unstratified sense). My attempt at a proof that Con(ZFC) implies Con(NFU + stuff above) works perfectly well to show that Con(ZFC + weakly compact cardinal) implies Con(NFU + stuff above). A variation I attempted turned out to need a refinement of the Erdos-Rado theorem which works -- in the presence of weakly compact cardinals! I find it very hard to believe that the Axiom of Cantorian Sets is that strong! Internal evidence seems to support strength of ZFC. The Axiom of Cantorian Sets is very useful for establishing that limit ordinals are strongly Cantorian; limit ordinals familiar from ZFC will generally be Cantorian, thus strongly Cantorian by the axiom. In my paper I proved that the Axiom of Cantorian Sets implies that "each class of strongly Cantorian isomorphism types of wfexts which is definable from equality and the "membership" relation natural for these objects with all quantifiers restricted to the class of strongly Cantorian isomorphism types of wfexts is the intersection of the class of strongly Cantorian isomorphisms of wfexts and some set". This principle is enough for the interpretation of Zermelo style set theory in s.c. isomorphism types of wfexts to satisfy Replacement. The idea of the proof of the preceding result I outlined during my visit. Let's call "ismorphism types of wfexts" "Z-pictures" (I have just created this term) (actually, we want only those types which belong to ranks all subsets of which are realized as types to be Z-pictures). Any class of Z-pictures defined as above has no problem with stratification except that quantifiers may be present which range over a non-set (the class of s.c. Z-pictures). Relativize all quantifiers to the lowest rank of Z-pictures which has the same theory as the class of all Z-pictures; this rank is Cantorian (obviously) and so strongly Cantorian. The rank to which one relativizes may be dependent on parameters in the definition of the class; as long as these parameters are s.c., everything works fine. When one carries out this process repeatedly, one needs to relativize to lowest sequences of ranks b1,...bn which have the same theory (considered as models of TTU_{n+1}) as Z (the last rank, the set of all Z-pictures), Tz,...,T^n{z}; again, this is a sequence of Cantorian, thus strongly Cantorian ranks. The reason that one needs iterated images under T is that the relativization process itself needs to be relativized, and the only way to do this is to reflect things downward using T (theories being preserved because Counting holds). In this way, all class definitions of this kind can be transformed into stratified definitions, as long as all parameters are restricted to s.c. objects; these stratified definitions define sets, which will have the same s.c. elements as the original classes. The class definition principle above is a weaker version of my Axiom of Small Ordinals. Showing that it implies Replacement is pretty easy. If NFU + Infinity + Choice + Axiom of Cantorian Sets _did_ turn out to be quite strong this would recommend it in some ways (the Axiom of Cantorian Sets is, after all, a very natural assumption). The strength of this axiom is in any event below that of a measurable cardinal, and probably below that of a Weakly compact cardinal by my busted argument. I will be surprised if Solovay can show that it is stronger than ZFC, but I must admit to being stymied in my efforts to show its exact strength is that of ZFC! My original argument does not seem to be possible to repair in any cheap way; all obvious variations seem to need that blasted weakly compact cardinal! --Randall From solovay@math.berkeley.edu Fri Apr 21 00:06 MDT 1995 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA09504; Fri, 21 Apr 1995 00:06:43 -0600 Return-Path: Received: by math.berkeley.edu (8.6.10/1.33(math)Ow.1) id XAA08437; Thu, 20 Apr 1995 23:02:12 -0700 Date: Thu, 20 Apr 1995 23:02:12 -0700 From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199504210602.XAA08437@math.berkeley.edu> To: holmes@catseye.idbsu.edu Subject: Addendum to letter 3 Lower Bounds On Consistency Strength Status: RO Randall, I clearly was rather tired when I wrote that letter since I forgot the main point. Work in NFUA. Recall that we have an ordering of V, <, which is "as short as possible". In terms of this we define an analogue of the T map as follows. Let x in V. Let S_x be the set of y such that y < x. We let S_x' be the set of singletons of members of S_x, and give it the isomorphic copy of the restriction of < to S_x as its ordering. So S_x' is order-isomorphic to S_y for a unique y, and we call that y TT(x). [I am using TT since this is not quite the T of your paper, though it is very closely related to it. Now go to the Specker framework described in my letter 3. The map TT has the following relationship to j. Let zeta be an ordinal of L_Omega. Then zeta has a copy in V_0, [which lies in Omega_0, in fact] and as described in that letter, there is a canonical injection of Omega_0 onto a proper initial segment of Omega_1 which is a subset of V_1. Say zeta* is the image of zeta under this map. Then j^{-1}(zeta*) = TT(zeta). This was difficult for me to prove, but I suspect that it is obvious to you, and so omit the proof. [I will, of course, supply my proof if need be.] What I need is the corollary. The L_Omega cardinal gamma is Cantorian iff j(gamma) = gamma. Hence gamma is strongly Cantorian iff the restriction of j to the ordinals less than gamma is the identity. From solovay@math.berkeley.edu Fri Apr 21 00:55 MDT 1995 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA09509; Fri, 21 Apr 1995 00:55:46 -0600 Return-Path: Received: by math.berkeley.edu (8.6.10/1.33(math)Ow.1) id XAA08937; Thu, 20 Apr 1995 23:51:16 -0700 Date: Thu, 20 Apr 1995 23:51:16 -0700 From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199504210651.XAA08937@math.berkeley.edu> To: holmes@catseye.idbsu.edu Subject: Letter 4: Lower bounds on consistency Status: RO We work in the Specker framework. The following definitions will take place within the model L_{Omega_2}. If I were formal, there would be a definition of the sequences I am constructing in L_Omega_2 from the parameters Omega_0 and Omega_1. Since L_Omega_2 is a model of either ZF- or Z, it has a perfectly good treatment of finite sequences of ordinals [the usual set-theoretic treatment!]. We are going to define, by induction on the integer i, ordinals alpha(i) and beta(i). It will be true that if alpha(i+1) is defined, then alpha(i+1) < alpha(i). Similarly, beta(i+1) < beta(i) if both are defined. It will always be true that if alpha(i) is defined and j is less than i, then alpha(j) is defined. So alpha is defined on a proper initial segment of the integers. Entirely analogous remarks apply to beta. It will also be true that alpha(i) is defined iff beta(i) is defined. To start the process off, alpha(0) = Omega_0; beta(0) = Omega_1. Now suppose that alpha(i) and beta(i) have been defined. Our task is to tell whether alpha(i+1) and beta(i+1) are defined and, if so, what they are. First, if alpha(i) = beta(i), then alpha(i+1) and beta(i+1) are undefined. Next, if alpha(i) and beta(i) have different "colors" [in a sense to be explained in a moment] then alpha(i+1) and beta(i+1) are undefined. Given an ordinal of Omega_2, we define its color as follows: 0) Ordinals <= omega get the color 0. 1) Ordinals that are not cardinals [in L_{Omega_2] get the color 1. [I won't keep repeating "in L_{Omega_2} in the remainder of this definition.] 2) Ordinals that are infinite sucessor cardinals get the color 2. 3) Singular limit cardinals get the color 3. 4) Inaccessible cardinals that are not limits of inaccessible cardinals get the color 4. 5) Inaccessible limits of inaccessible's get the color 5. The only remaining cases to consider are when alpha(i) and beta(i) have the same color, but are distinct ordinals. There are six subcases according to what the common color is. Case 0: alpha(i+1) and beta(i+1) are undefined in this case. Case 1: Set alpha(i+1) equal to the cardinal of alpha(i) in L_{Omega_2}. Set beta(i+1) equal to the cardinal of beta(i) in L_{Omega_2}. Case 2: Set alpha(i+1) equal to the largest cardinal less than alpha(i). Define beta(i+1) analogously from beta(i). Case 3: This is the most complicated case to handle. Let alpha* be the cofinality of alpha; let beta* be the cofinality of beta. If alpha* is unequal to beta*, set alpha(i+1) = alpha* and beta(i+1) = beta*. So suppose now that alpha* = beta* = gamma, say. Let f be the L-least order preserving map of gamma cofinally into alpha(i) and let g be the L-least order preserving map of gamma cofinally into beta(i). Let eta be the least ordinal less than gamma such that f(eta) is unequal to g(eta). [eta must exist since alpha(i) is unequal to beta(i).] Set alpha(i+1) = f(eta); set beta(i+1) = g(eta). Case 4: Let alpha(i+1) be the sup of the inaccessible cardinals less than alpha(i). [If there are none, alpha(i+1) = 0.] Let beta(i+1) be defined analogously from beta(i). Case 5: In this case, the construction stops and alpha(i+1) and beta(i+1) are undefined. This completes the construction and letter 4. From T.Forster@pmms.cam.ac.uk Fri Apr 21 02:20 MDT 1995 Received: from diamond.idbsu.edu by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA09516; Fri, 21 Apr 1995 02:20:44 -0600 Return-Path: Received: from emu.pmms.cam.ac.uk by diamond.idbsu.edu with SMTP (1.37.109.16/16.2) id AA069752366; Fri, 21 Apr 1995 02:19:26 -0600 Received: by emu.pmms.cam.ac.uk (UK-Smail 3.1.25.1/1); Fri, 21 Apr 95 08:50 BST Message-Id: Date: Fri, 21 Apr 95 09:15 BST From: Thomas Forster To: holmes@diamond.idbsu.edu Subject: every can is stcan Status: RO Thanks for bringing me up to date with your tho'rts on this. I haven't actually tho'rt about this myself for many years, and the of course in the context of NF not NFU. I think my view was that it was a very unsatisfactory axiom, having possibly rather strong consequences for big sets and not saying as much about little sets as one would like. It doesn't seem to imply that the Hcan sets are a model of ZF for example, which is something one might want. On the other hand i have the feeling that funny things happen to big ordinals if every can is stcan. I think i ended up feeling that one should steer clear of it. I suppose what i want to say is that i don't yet think i understand the genesis of the two concepts of can and stcan to know whther or not they should have the same extensions, and certainly *at the moment* i can see no motivation. But keep keeping me up to date! best wishes Thomas From T.Forster@pmms.cam.ac.uk Fri Apr 21 07:47 MDT 1995 Received: from emu.pmms.cam.ac.uk by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA09585; Fri, 21 Apr 1995 07:47:02 -0600 Return-Path: Received: by emu.pmms.cam.ac.uk (UK-Smail 3.1.25.1/1); Fri, 21 Apr 95 14:39 BST Message-Id: Date: Fri, 21 Apr 95 14:39 BST From: Thomas Forster To: holmes@catseye.idbsu.edu Subject: Re: every can is stcan Status: RO Really? How do you get all the comprehension? Don't you need a certain amount of choice? Oh yes, i forgot, you are doing all this in \nf U. Well, *that* does it! From holmes@catseye.idbsu.edu Fri Apr 21 09:08 MDT 1995 Received: by catseye.idbsu.edu (1.38.193.4/16.2) id AA09683; Fri, 21 Apr 1995 09:08:40 -0600 Date: Fri, 21 Apr 1995 09:08:40 -0600 From: Randall Holmes Return-Path: To: holmes@catseye.idbsu.edu, solovay@math.berkeley.edu Subject: Re: Addendum to letter 3 Lower Bounds On Consistency Strength Status: RO Dear Bob, I was too quick in confirming your statements about your map TT(x) (I didn't read the definition carefully enough!). I think that it may very well be correct, but I _do_ need to think about it and may request the proof! --Randall From holmes@catseye.idbsu.edu Fri Apr 21 09:15 MDT 1995 Received: by catseye.idbsu.edu (1.38.193.4/16.2) id AA09706; Fri, 21 Apr 1995 09:10:54 -0600 Date: Fri, 21 Apr 1995 09:10:54 -0600 From: Randall Holmes Return-Path: To: T.Forster@pmms.cam.ac.uk, holmes@catseye.idbsu.edu Subject: Re: every can is stcan Status: RO I don't think that choice is needed in the argument, actually, but if it were, it would be available. Remember that Z-pictures have structure on them which the whole universe doesn't. --Randall From solovay@math.berkeley.edu Fri Apr 21 00:55 MDT 1995 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA09509; Fri, 21 Apr 1995 00:55:46 -0600 Return-Path: Received: by math.berkeley.edu (8.6.10/1.33(math)Ow.1) id XAA08937; Thu, 20 Apr 1995 23:51:16 -0700 Date: Thu, 20 Apr 1995 23:51:16 -0700 From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199504210651.XAA08937@math.berkeley.edu> To: holmes@catseye.idbsu.edu Subject: Letter 4: Lower bounds on consistency Status: RO We work in the Specker framework. The following definitions will take place within the model L_{Omega_2}. If I were formal, there would be a definition of the sequences I am constructing in L_Omega_2 from the parameters Omega_0 and Omega_1. Since L_Omega_2 is a model of either ZF- or Z, it has a perfectly good treatment of finite sequences of ordinals [the usual set-theoretic treatment!]. We are going to define, by induction on the integer i, ordinals alpha(i) and beta(i). It will be true that if alpha(i+1) is defined, then alpha(i+1) < alpha(i). Similarly, beta(i+1) < beta(i) if both are defined. It will always be true that if alpha(i) is defined and j is less than i, then alpha(j) is defined. So alpha is defined on a proper initial segment of the integers. Entirely analogous remarks apply to beta. It will also be true that alpha(i) is defined iff beta(i) is defined. To start the process off, alpha(0) = Omega_0; beta(0) = Omega_1. Now suppose that alpha(i) and beta(i) have been defined. Our task is to tell whether alpha(i+1) and beta(i+1) are defined and, if so, what they are. First, if alpha(i) = beta(i), then alpha(i+1) and beta(i+1) are undefined. Next, if alpha(i) and beta(i) have different "colors" [in a sense to be explained in a moment] then alpha(i+1) and beta(i+1) are undefined. Given an ordinal of Omega_2, we define its color as follows: 0) Ordinals <= omega get the color 0. 1) Ordinals that are not cardinals [in L_{Omega_2] get the color 1. [I won't keep repeating "in L_{Omega_2} in the remainder of this definition.] 2) Ordinals that are infinite sucessor cardinals get the color 2. 3) Singular limit cardinals get the color 3. 4) Inaccessible cardinals that are not limits of inaccessible cardinals get the color 4. 5) Inaccessible limits of inaccessible's get the color 5. The only remaining cases to consider are when alpha(i) and beta(i) have the same color, but are distinct ordinals. There are six subcases according to what the common color is. Case 0: alpha(i+1) and beta(i+1) are undefined in this case. Case 1: Set alpha(i+1) equal to the cardinal of alpha(i) in L_{Omega_2}. Set beta(i+1) equal to the cardinal of beta(i) in L_{Omega_2}. Case 2: Set alpha(i+1) equal to the largest cardinal less than alpha(i). Define beta(i+1) analogously from beta(i). Case 3: This is the most complicated case to handle. Let alpha* be the cofinality of alpha; let beta* be the cofinality of beta. If alpha* is unequal to beta*, set alpha(i+1) = alpha* and beta(i+1) = beta*. So suppose now that alpha* = beta* = gamma, say. Let f be the L-least order preserving map of gamma cofinally into alpha(i) and let g be the L-least order preserving map of gamma cofinally into beta(i). Let eta be the least ordinal less than gamma such that f(eta) is unequal to g(eta). [eta must exist since alpha(i) is unequal to beta(i).] Set alpha(i+1) = f(eta); set beta(i+1) = g(eta). Case 4: Let alpha(i+1) be the sup of the inaccessible cardinals less than alpha(i). [If there are none, alpha(i+1) = 0.] Let beta(i+1) be defined analogously from beta(i). Case 5: In this case, the construction stops and alpha(i+1) and beta(i+1) are undefined. This completes the construction and letter 4. From holmes Mon Apr 17 14:11:48 1995 To: solovay@math.berkeley.edu Subject: Construction 3 Status: RO Dear Bob, Now, hopefully, you are talking to the real Randall Holmes with brain completely engaged :-) Topic 1: Modelling counting (not yet NFUA!) I am going to describe the basic construction of a model of NFU with a certain standard infinite cardinal \beta (you may think of it as \omega, so that this will be a model of Counting, but I will be general, since generality is cheap here) strongly cantorian. It is sufficient to build a model with automorphism in which each ordinal less than \beta (and \beta itself) is fixed by the automorphism. Choose a cardinal \alpha greater than each iterated exponential of \beta (so that the Erdos-Rado theorem can be used). Functions from \alpha to \beta can be coded by elements of \alpha; clearly, so can finite sets of such functions. Consider, for each function f from \alpha to \beta, the set C_f of all codes of finite sets of maps from \alpha to \beta which contain f. The collection of C_f's is a filter on \alpha which can be extended to an ultrafilter C. C "describes" the code of a "finite set" of maps \alpha -> \beta which contains all standard such maps, in the ultrafilter model constructed using C. This "finite set" of functions induces a partition of \alpha into \beta pieces, which has a "homogeneous set" H. We use this "homogeneous set" to define a sequence of ultrafilters U_i on [\alpha]^i in the real world: U_i will contain all standard finite sets of subsets of \alpha of size i whose analogues in the model built with C contain all subsets of H with i elements. The model can be defined in terms of the U_i's as in the earlier constructions. The model contains \beta_k's indexed by integers k: U_i tells us which sets contain each sequence of i successive \beta_k's; the model elements are exactly the images under standard functions of finite sequences of successive \beta_k's. Any ordinal less than \beta in this model is the image of a finite sequence of n successive \beta_k's under some standard function \alpha^n -> \beta. Now the set H in the model built using C was homogeneous with respect to the partition induced by this map (and every similar standard map), so the images of any sequence of n successive \beta_k's remains the same when the indexes of the \beta_k's are incremented; any ordinal less than \beta is fixed by the automorphism. If you have trouble seeing that this information is coded in the U_i's, consider that the information that f(beta_0...beta_{n-1}) = f(beta_1...beta_n) for any standard f and n is coded in U_{n+1}. (I assume that you will _not_ have trouble seeing this; I include it mostly to remind _myself_). This construction gives models of Counting and higher axioms of the form "Ordinal so-and-so is strongly Cantorian". It does not, of course, give us a basis for any assertion about _all_ Cantorian or s.c. sets. I seem not to have included an argument for a model of NFUA using an inaccessible in the paper as it stands; I'm going to have to reconstruct this! It was a refinement of this argument for models with s.c. sets, and I seem to have thought that it was redundant given the sharper result for ZFC which I have not communicated successfully! I will look at old drafts of the paper and see if I can find that construction; meanwhile, do you believe this construction? Topic 2: A possible description of the set of sentences needed to establish that Con(ZFC) = Con(NFUA) As in the argument in the paper, we work in a term model of ZFC. Add alpha_i's, nonstandard ordinals, indexed by integers (levels can be recovered as V_{\alpha_i's}) They satisfy the following sentences: G(\alpha_1...\alpha_n) < \alpha_{n+1}, G standard For each partition P of finite subsets of the ordinals of size n into set many compartments, definable using a finite sequence of \alpha_i's whose largest index is k, all \alpha_i's with index higher than k belong to the same cofinal homogeneous class for P. The question about the latter collection of sentences (in my mind) is again whether it is expressible without essential reference to classes. I'll take another stab at expressing it: Let F(\alpha_{k-m}...\alpha_k) be a class map from n-element subsets (represented as ascending sequences) of the ordinals onto some (von Neumann) ordinal (i.e., it has bounded values, and so is a partition of the kind indicated above). The part of the definition of this partition represented by the letter F is standard (in the term model). Then we have F(\alpha_{j_1}...\alpha_{j_n}) = F(\alpha_{k_1}...\alpha_{k_n}) whenever all j_m's, k_m's > k. I haven't directly provided that the homogeneous class is cofinal (which is essential for iteration; one doesn't want it to become small!), but this seems to be provided by the first set of sentences; any class which contains all the \alpha_i's seems to have to be cofinal. But this condition seems to be expressible as well (without essential reference to classes): to say that a given homogeneous class can be regarded as cofinal is to say that the corresponding value for F is found on homogeneous sets relative to this partition of arbitrarily large ordinals, I think? Do you agree that this set of sentences is describable and consistent (I do say something like this in the paper)? Existence of cofinal homogeneous classes does follow from Erdos-Rado, does it not? It appears that this set of sentences is sufficient for my purposes (the "standard-bounded" stuff is not needed). Compare any \alpha_i with a standard function image F(\alpha_{i+1}...\alpha_{i+n}). Consider the partition produced by mapping each sequence (a_1...a_n) to F(a_1...a_n) if this is less than \alpha_i and to \alpha_i otherwise. The larger \alpha_k's must be homogeneous with respect to this partition: thus, F(\alpha_{i+1}...\alpha_{i+n}) is either (i.) greater than or equal to \alpha_i or (ii.) lies below \alpha_i and is fixed under the automorphism induced by incrementing the indices of the \alpha_i's (that this operation is an automorphism follows from the homogeneity properties of the model). Thus, in the induced model of NFU (built from V_{\alpha_0) using the index incrementing automorphism), every ordinal will either be fixed under the automorphism or greater than some \alpha_i; this is sufficient for the Axiom of Cantorian Sets to hold. Final Remark: Now that I have what I think is the correct set of sentences for the Con(ZFC) -> Con(NFUA) proof, I think I see how to reconstruct the model of NFUA below an inaccessible. But I will get this part out before tackling the latter in the next message! I hope that I am being more coherent! --Randall From solovay@math.berkeley.edu Sat Apr 22 00:35 MDT 1995 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA10923; Sat, 22 Apr 1995 00:35:01 -0600 Return-Path: Received: by math.berkeley.edu (8.6.10/1.33(math)Ow.1) id XAA03663; Fri, 21 Apr 1995 23:30:29 -0700 Date: Fri, 21 Apr 1995 23:30:29 -0700 From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199504220630.XAA03663@math.berkeley.edu> To: holmes@catseye.idbsu.edu Subject: The final installment of the proof Status: RO Randall, I decided to send this, even though you seem to accept the proof after installments 1 through 4, since there are some delicate points I have not yet discussed which were not reflected in your comments. We have defined two elements of L_{Omega_2}, alpha and beta. They are finite decreasing sequences of ordinals of the same length. We wish to prove by induction on i in omega the following claims: (0) alpha(i) and beta(i) are defined; (1) alpha(i) is unequal to beta(i); (2) j(alpha(i)) = beta(i). The set of i satisfying (0) or (1) is clearly a set of our Specker model. For (2), this is less clear because of the appearance of j. We get around this as follows. Let gamma be the sequence j(alpha). Then gamma is a finite sequence on L_{Omega_3} which is strictly decreasing and starts with Omega_1. It follows that gamma is in L_{Omega_2}. Next, using the axiom of counting, we see that the length of gamma is j(m) = m, where m is the common length of alpha and beta. Moreover, gamma(i) = gamma(j(i)) = j(alpha(i)). Thus we can describe the set of i satisfying (2) alternatively as the set of i such that beta(i) = gamma(i). This alternative decription makes evident that the set of i satisfying (2) lies in our Specker model. [In fact, since its finite, it also lies in L_{Omega_2}, but this isn't terribly important.] Clearly 0 satisfies (0) through (2). Let i be maximal which satisfies (0) through (2). We show that i+1 satisfies (0) through (2) as well, which will give our desired contradiction. Since i satisfies (1), alpha(i) is unequal to beta(i). Since j(alpha(i) = beta(i), we conclude easily that alpha(i) and beta(i) have the same color. The remainder of the argument splits into cases according to the common color of alpha(i) and beta(i). Case 0. Then alpha(i) <= omega. So, by the axiom of counting, j(alpha(i)) = alpha(i). So alpha(i) = beta(i). This contradicts that i satisfies (1). So this case can't arise. Case 1. Then clearly alpha(i+1) and beta(i+1) are defined and j(alpha(i+1)) = beta(i+1). We have to rule out that alpha(i+1) = beta(i+1). But if this happens, alpha(i+1) is Cantorian. Hence the successor cardinal to alpha(i+1) is also Cantorian [say using the characterization of fixed points of j.} So it is strongly Cantorian. But alpha(i) is less than this successor, so it is cantorian. So beta(i) = j(alpha(i)) = alpha(i), which contradicts our inductive hypothesis. Case 2: Quite similar to Case 1. The details are omitted. Case 3: It is this case where we make the most crucial use of the axiom that every cantorian ordinal is strongly cantorian. There are two subcases. (A) Suppose alpha(i+1) = cf(alpha(i)) is unequal to cf(beta(i)). Then all our inductive claims are clear. (B) Suppose that the cofinalities of alpha(i) and beta(i) are both equal to gamma, say. Then j(gamma) = j(cf(alpha(i))= cf(j(alpha(i)))= cf(beta(i)) = gamma. [We are using constantly that j is an elementary embedding from L_{Omega_1} to L_{Omega_2} and that things like cardinality and cofinality are absolute from L_{Omega_1} to L_{Omega_2} since Omega_1 is a cardinal in L_{Omega_2}. The upshot is that j(gamma) = gamma. Let eta be least such that f(eta) is unequal to g(eta). [Definitions as in the treatment of this case in the prior letter. By the Cantorian axiom, j(eta) = eta. Clearly, j(f) = g. So j(alpha(i+1)) = j(f(eta)) = j(f)(j(eta)) = g(eta) = beta(i+1). So all inductive claims are clear. Case 4. This case is easy and similar to Cases 1 and 2. If alpha(i+1) = beta(i+1), then alpha(i) = beta (i) = least inaccessible greater than alpha(i+1), contradicting our inductive hypothesis. Case 5. By the assumption that the main claim is false, there is only one ordinal of color 5. So this case can't arise. That's all folks. From solovay@math.berkeley.edu Sat Apr 22 00:43 MDT 1995 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA10928; Sat, 22 Apr 1995 00:43:56 -0600 Return-Path: Received: by math.berkeley.edu (8.6.10/1.33(math)Ow.1) id XAA03815; Fri, 21 Apr 1995 23:39:24 -0700 Date: Fri, 21 Apr 1995 23:39:24 -0700 From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199504220639.XAA03815@math.berkeley.edu> To: holmes@catseye.idbsu.edu In-Reply-To: Randall Holmes's message of Fri, 21 Apr 1995 12:43:45 -0600 <199504211839.LAA19459@math.berkeley.edu> Subject: A question Status: RO Randall, You ask: Do you have any sense for exactly how strong NFUA should be? Now that I see what I was doing wrong, I think that I can show easily that the existence of a weakly compact cardinal implies Con(NFUA); the broken link in my best argument involves cofinal homogeneous sets for partitions, and below a weakly compact cardinal, these can be found. Reply: I have mentioned what I can do re this in previous letters. To sum up: The weakest large cardinal property from which I can get models of NFUA is what I dubbed Phi in a prior letter. This is much stronger than "weakly compact". For example, it implies the esistence of a weakly compact cardinal such that the set of smaller weakly compact cardinals is stationary in it. I can improve my argument in various trivial ways. But I can't deduce the consistency of a Mahlo cardinal from that of NFUA. The gap between these upper and lower bounds is enormous! I am rather sceptical you can get a model of NFUA from a weakly compact. I think you have still not understood all the bugs in your prior proofs. [But I could be wrong about this. I certainly don't know that Exists weakly compact implies Con(NFUA) is not a theorem of ZFC.] As ever, Bob From solovay@math.berkeley.edu Sun Apr 23 12:21 MDT 1995 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA11716; Sun, 23 Apr 1995 12:20:59 -0600 Return-Path: Received: by math.berkeley.edu (8.6.10/1.33(math)Ow.1) id LAA23547; Sun, 23 Apr 1995 11:16:27 -0700 Date: Sun, 23 Apr 1995 11:16:27 -0700 From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199504231816.LAA23547@math.berkeley.edu> To: holmes@catseye.idbsu.edu In-Reply-To: Randall Holmes's message of Sun, 23 Apr 1995 11:28:53 -0600 <199504231724.KAA23035@math.berkeley.edu> Subject: Further stuff Status: RO Dear Randall, I had been planning [and still am] to read your construction 3 and the proof of the consistency of NFUA given a weakly compact. I am even willing to read your proof of Con(NFUA) --> Con(ZFC). But this list may be "it". Weakly compact cardinals are just the Pi^1_1 indescribable cardinals. This suggests immediately what the analogue of ZFC should be [it will be in a language with both sets and classes sort of like Kelley-Morse set theory]. Today is committed to other tasks so I can't be more explicit at the instant. You may state and prove my theorem duly attributed. I would like it then to be in a self-contained appendix which I would then proofreand and approve. I have a real horror about having arguments of mine being first presented to the world with incorrect proofs [as has happened at least once] so I do insist on approving the presentation. I think the letters I sent go a long way to indicating how I would want it presented, but I would certainly rewrite more carefully Letter 3, and a proof of the claim in the appendix to Letter 3 should be provided. As ever, Bob From holmes Mon Apr 24 12:14:30 1995 To: solovay@math.berkeley.edu Subject: Construction 3 (proofread) Status: RO Dear Bob, This is the proofread version of the latest edition of construction 3. Definitely don't read the version in the paper; I started by misstating the domain of the first ultrafilter (it certainly cannot be alpha; it needs to be something like 2^{2^alpha}} as below (as I knew perfectly well on some level...) --Randall Detailed account of construction 3 Construction: Build a model of NFU in which a fixed ordinal beta is strongly Cantorian. Let alpha be the first strong limit cardinal greater than beta. Let A_m be the set of maps [alpha]^m -> beta for each natural number m. Let A = the union of the A_m's. Let U be an ultrafilter on the power set of A which contains {x|x is a finite subset of A} and contains {x|a \in x} for each a \in A. Build an ultrafilter model of set theory using U in the standard way; then the equivalence class of the identity function will be a nonstandard finite set F which contains all (nonstandard analogues of) standard partitions of [alpha]^n for standard n into beta pieces (in the sense of the model). In this model, we can build a single partition of [alpha]^M for a nonstandard natural number M using the set F which codes all of the partitions in F. Suppose that the elements of F are indexed: {F_1,...,F_N}. Each F_i has domain [alpha]^{m_i}; let M = max(m_i) and define F_i' as the map with domain [alpha]^M obtained by applying F_i to the initial segment of length m_i of elements of [alpha]^M. We can define a map P0: [alpha]^M -> beta^N (recall that N = card(F)): the ith projection of P0(a_1,...,a_m,...,a_M) is F_i'(a_1,...,a_m). Compose P0 with any injection from beta^N into beta to obtain the desired partition P: [alpha]^M -> beta. By the Erdos-Rado theorem, P has a homogeneous set H. It should be clear that this homogeneous set H will be homogeneous for the nonstandard analogue of each standard partition of an [alpha]^n into beta pieces. We now define a sequence of ultrafilters U_i indexed by positive integers in the real world (not in the nonstandard model!). U_i will be the set of all subsets of [alpha]^i whose analogues in the nonstandard model built with U above contain a tuple (a_1,...,a_i) of ascending distinct elements of H. Clearly, if (the analogue of) a standard set contains any one of these, it contains all of them! We use the sequence of ultrafilters U_i to build a second nonstandard model of set theory. An ultrafilter model built with a U_n will contain a sequence of n indiscernible nonstandard elements a_1...a_n of alpha (ordinals below alpha). Every object in this model will be of the form F(a_i,...,a_i+k) where F is a standard function [alpha]^k -> V_alpha and the arguments are a segment of the sequence of a_i's. The full nonstandard model is obtained as a direct limit of ultrafilter models U_{2n+1} with the identification of the indiscernibles in successive models indicated by calling each sequence of 2n+1 indiscernibles (a^{-n},...,a_n); the direct limit contains a collection of indiscernibles a_i indexed by the integers. It should be clear that the direct limit has an external automorphism induced by incrementing the indexes of the a_i's uniformly. Further, we observe that any element (standard or nonstandard) of the analogue of beta in this model is fixed under the automorphism. For any such element may be represented without loss of generality by a term F(a_i,...a_i+k), where F is a standard function from [alpha]^i to beta, and the model construction ensures that the set of all sequences b_1,...,b_k+1 such that F(b_1,...,b_k) = F(b_2,...,b_{k+1}) belongs to U_{k+1} for any standard F:[alpha_i] -> beta, which ensures in turn that the general element of beta that we have described is fixed by the automorphism. This is sufficient to guarantee that the analogue of beta in this model is strongly Cantorian in the natural model of NFU with domain V_{a_0} constructed inside the nonstandard model of V_alpha derived from the U_i's. Refinement of Construction 3 which yields a model of NFUA below a weakly compact cardinal: Let kappa be a weakly compact cardinal. We build an ultrafilter U on a suitable set which codes a finite set F containing all standard maps [kappa]^m -> kappa for standard n. We work in an ultrafilter model of set theory containing this object. For each alpha < kappa (including nonstandard ones!), define F_alpha as the set of maps obtained by composing elements of F with the map which is the identity on ordinals less than alpha and sends each ordinal >= alpha to alpha. F_alpha codes all standard partitions (relative to alpha, which may itself be nonstandard) of [kappa]^n into pieces indexed by the ordinals <= alpha. Just as above, we can define a partition P_alpha of a [kappa]^{M_alpha} which codes all partitions standard relative to alpha. We choose a sequence of ordinals alpha_i of the nonstandard model as follows: let alpha_{0} = 1. We define H_{0} as a homogeneous set of cardinality kappa for P_1. We choose each alpha_{i+1} from H_{i} and define H_{i+1} as a homogeneous set of cardinality kappa for the partition P_{alpha_{i+1}} restricted to H_{i} (which is large enough; this was the point of failure of my construction below an inaccessible!). The ultrafilters U_i are defined as the collections of standard subsets of [kappa]^i which contain the sequence of the first consecutive alpha_i's starting with a_1 (not a_0!). The second model is constructed exactly as above as a directed limit of models built using U_{2n+1}'s, in which there will be a collection of indiscernibles indexed by the integers. The objects described by the ultrafilters now have the property that if F(alpha_i+1,...alpha_i+j) < alpha_i, it is fixed by the automorphism (think about the partition induced by F_{alpha_i}; all a_k's above a_i belong to a homogeneous set with respect to this partition; in fact, F(alpha_i+1,...,a_i+j) is forced to lie below all a_k's!). Any object in the model of NFU based on V_{alpha_0} has rank either between successive alpha_i's (and so not Cantorian) or below all alpha_i's (and so, by the remarks above, strongly Cantorian, because fixed by the automorphism along with everything below it). There should be an analogous argument for Con(NFUA) from Con(the theory of the part of the universe below a weakly compact cardinal)? --Randall From solovay@math.berkeley.edu Wed Apr 26 09:41 MDT 1995 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA15445; Wed, 26 Apr 1995 09:41:33 -0600 Return-Path: Received: by math.berkeley.edu (8.6.10/1.33(math)Ow.1) id IAA25943; Wed, 26 Apr 1995 08:36:58 -0700 Date: Wed, 26 Apr 1995 08:36:58 -0700 From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199504261536.IAA25943@math.berkeley.edu> To: holmes@catseye.idbsu.edu Subject: I now believe Status: RO Randall, Thanks to your latest letter I now believe the proof of (a) construction of a model of the axiom of counting via Construction 3; (b) getting a model of NFUA from a weakly compact cardinal. My actual process of verification consisted of (a) reading your proofs, definitely without full comprehension and then (b) going off and hacking out proofs of my own. My proof of the "counting" result is virtually identical with yours, but my proof of the weakly compact result is somewhat different [and I think slightly simpler]. One result of this method of proceeding is that I can't vouch for your proofs line by line. Of course, I now see that various comments I made about the likeliness of these results and proofs being correct were flat out wrong. To paraphrase Judge Ito, "That's life in the big city.". The upper bound and lower bound on the consistency strength of NFUA are starting to get awfully close. I will probably take a shot again at getting a Mahlo in L from NFUA. What I think should be easier [but have not yet done] is find a large cardinal axiom [probably very ad-hoc] strictly weaker than weakly compact cardinals and which yields the consistency of NFUA. Another project which looks viable to me is to show now that one can get an inaccessible limit of weakly compacts from NFUB. [I certainly don't have a clear idea as to how to do this yet.] This would settle that the two theories are of strictly different consistency strengths. As ever, Bob From holmes@catseye.idbsu.edu Wed Apr 26 10:17 MDT 1995 Received: by catseye.idbsu.edu (1.38.193.4/16.2) id AA15458; Wed, 26 Apr 1995 10:17:24 -0600 Date: Wed, 26 Apr 1995 10:17:24 -0600 From: Randall Holmes Return-Path: To: holmes@catseye.idbsu.edu, solovay@math.berkeley.edu Subject: Re: I now believe Status: RO I was thinking about the definition of a property of cardinals: Let kappa be an "NFUA cardinal" if it has the following property: If f is a map from [kappa]^n -> alpha, alpha < kappa, then there is an increasing sequence {x_i} of elements of kappa, indexed by the natural numbers, such that the image of any set of n _consecutive_ terms of the sequence under f is the same. Is an "NFUA cardinal" weakly compact, or is this a weaker property? It is tailored exactly to prove the existence of models of NFUA, of course! --Randall From holmes@catseye.idbsu.edu Wed Apr 26 10:20 MDT 1995 Received: by catseye.idbsu.edu (1.38.193.4/16.2) id AA15466; Wed, 26 Apr 1995 10:20:58 -0600 Date: Wed, 26 Apr 1995 10:20:58 -0600 From: Randall Holmes Return-Path: To: holmes@catseye.idbsu.edu, solovay@math.berkeley.edu Subject: Re: I now believe Status: RO Another point: I think that the strength of Jensen's proof of the consistency of Counting is misleading, because he proved counting by producing an omega-model, and the (inexpressible in first-order terms) assertion "All natural numbers are standard" is _much_ stronger than the Axiom of Counting; it implies full unstratified math induction for example, which is stronger than Counting! I'm glad that I had _something_ right :-) --Randall From holmes@catseye.idbsu.edu Wed Apr 26 10:22 MDT 1995 Received: by catseye.idbsu.edu (1.38.193.4/16.2) id AA15469; Wed, 26 Apr 1995 10:22:49 -0600 Date: Wed, 26 Apr 1995 10:22:49 -0600 From: Randall Holmes Return-Path: To: holmes@catseye.idbsu.edu, solovay@math.berkeley.edu Subject: Re: I now believe Status: RO A third remark: If you go on to read the Con(NFUA) -> Con(ZFC) construction, I'd like to write it up again; the main feature of this paper is that I am now severely dissatisfied with the way almost everything in it is written! The way it is written in the paper is too telegraphic, and misses some important points that need to be made! --Randall From solovay@math.berkeley.edu Wed Apr 26 18:18 MDT 1995 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA16111; Wed, 26 Apr 1995 18:18:49 -0600 Return-Path: Received: by math.berkeley.edu (8.6.10/1.33(math)Ow.1) id RAA11719; Wed, 26 Apr 1995 17:14:14 -0700 Date: Wed, 26 Apr 1995 17:14:14 -0700 From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199504270014.RAA11719@math.berkeley.edu> To: holmes@catseye.idbsu.edu Subject: Closing in for the kill Status: RO Randall, I'm glad to wait and read a revised writeup of Con(NFUA) -> Con(ZFC). Heres a theorem I can prove. Suppose that for every n, ZFC + exists an n-Mahlo is consistent. Then so is NFUA. I conjecture that this is sharp. That is for every n, I conjecture NFUA proves there is an n-Mahlo in L. Of course, this latest result supercedes your result that it there is a weakly compact, then NFUA is consistent. As ever, Bob From solovay@math.berkeley.edu Wed Apr 26 18:25 MDT 1995 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA16115; Wed, 26 Apr 1995 18:25:36 -0600 Return-Path: Received: by math.berkeley.edu (8.6.10/1.33(math)Ow.1) id RAA11893; Wed, 26 Apr 1995 17:21:01 -0700 Date: Wed, 26 Apr 1995 17:21:01 -0700 From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199504270021.RAA11893@math.berkeley.edu> To: holmes@catseye.idbsu.edu In-Reply-To: Randall Holmes's message of Wed, 26 Apr 1995 17:02:38 -0600 <199504262258.PAA09605@math.berkeley.edu> Subject: Homogeneous sets Status: RO Randall, You may well be right. I want to think about getting n-Mahlos from NFUA [which would settle things completely] before thinking about your question. It is of course still possible that the consistency strength of NFUA is very weak [less than a Mahlo], though at the moment I'm guessing the other way. As ever, Bob From solovay@math.berkeley.edu Wed Apr 26 23:34 MDT 1995 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA16312; Wed, 26 Apr 1995 23:34:04 -0600 Return-Path: Received: by math.berkeley.edu (8.6.10/1.33(math)Ow.1) id WAA17945; Wed, 26 Apr 1995 22:29:29 -0700 Date: Wed, 26 Apr 1995 22:29:29 -0700 From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199504270529.WAA17945@math.berkeley.edu> To: holmes@catseye.idbsu.edu Subject: the relevant reference Status: RO Journal of Symbolic Logic v. 37 (1972) On power-like models for hyperinaccessible cardinals,by James H. Schmerl and Saharon Shelah, pp. 531-537 Read sections 0 and 1; after understanding them, my result should be an easy exercise. [Alternatively, when you know this material, I can concisely explain my proof.] As ever, Bob From solovay@math.berkeley.edu Sun Apr 30 10:49 MDT 1995 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA18649; Sun, 30 Apr 1995 10:49:32 -0600 Return-Path: Received: by math.berkeley.edu (8.6.10/1.33(math)Ow.1) id JAA12989; Sun, 30 Apr 1995 09:44:54 -0700 Date: Sun, 30 Apr 1995 09:44:54 -0700 From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199504301644.JAA12989@math.berkeley.edu> To: holmes@catseye.idbsu.edu Subject: Radio silence Status: RO Randall, I hope my comments re the Shelah-Schmerl paper did not offend in some way. It really is true that one has to understand that paper to understand my improved upper bounds on the strength of NFUA. Very roughly, they show how from the stated large cardinal assumptions to construct a model of ZFC with something like a measure on the class of all ordinals. One can play the usual games with this "measure" getting indiscernibles and models of NFUA. The astonishing thing for me is that one gets a "measure" from such weak assumptions. Their proof uses compactness and one definitely does not get omega models. There is a companion paper of Schmerl in the Transactions of the AMS [the preciese reference is not at hand as I type] where he gives a combinatorial equivalent to n-Mahlo; I suspect that this will eventually yield the lower bounds, but that it will not be easy. [Despite comments to the contrary that I think I wrote, the paper itself is not terribly difficult to read or understand.] As ever, Bob From solovay@math.berkeley.edu Sun Apr 30 11:08 MDT 1995 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA18654; Sun, 30 Apr 1995 11:08:43 -0600 Return-Path: Received: by math.berkeley.edu (8.6.10/1.33(math)Ow.1) id KAA13134; Sun, 30 Apr 1995 10:04:05 -0700 Date: Sun, 30 Apr 1995 10:04:05 -0700 From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199504301704.KAA13134@math.berkeley.edu> To: holmes@catseye.idbsu.edu Subject: Missing reference Status: RO The Schmerl paper is in vol. 188 of the Transactions of the AMS (1974) on pages 281-291. --Bob From solovay@math.berkeley.edu Tue May 2 10:00 MDT 1995 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA20148; Tue, 2 May 1995 10:00:32 -0600 Return-Path: Received: by math.berkeley.edu (8.6.10/1.33(math)Ow.1) id IAA27878; Tue, 2 May 1995 08:55:51 -0700 Date: Tue, 2 May 1995 08:55:51 -0700 From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199505021555.IAA27878@math.berkeley.edu> To: holmes@catseye.idbsu.edu Subject: Package sent Status: RO A first class envelope [largeish] was sent yesterday with the two papers of Schmerl and the joint paper of Schmerl and Shelah. The joint paper is definitely relevant to the upper bound on the consistency of NFUA; of the two papers by Schmerl, I read through the one in the Transactions. At one time, I had hopes it would be directly relevant to the proofs of the corresponding lower bounds, but now I'm not so sure. My current thoughts are along the lines of ramping up the proof I sent you that one can get an inaccessible limit of inaccessibles in L, from NFUA. Alas, I must give a talk this Friday on a totally unrelated topic ["Quantum Turing Machines"], so I won't be spending much time on improved lower bounds for the consistency strength of NFUA in the next few days. It does seem now that this is significantly harder than the upper bounds, but problems often seem harder to me before they are solved. As ever, Bob From solovay@math.berkeley.edu Tue May 2 12:58 MDT 1995 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA20407; Tue, 2 May 1995 12:58:06 -0600 Return-Path: Received: by math.berkeley.edu (8.6.10/1.33(math)Ow.1) id LAA03754; Tue, 2 May 1995 11:53:20 -0700 Date: Tue, 2 May 1995 11:53:20 -0700 From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199505021853.LAA03754@math.berkeley.edu> To: holmes@catseye.idbsu.edu In-Reply-To: Randall Holmes's message of Tue, 2 May 1995 10:53:43 -0600 <199505021649.JAA29925@math.berkeley.edu> Subject: My sketch Status: RO Randall, I will add your sketch to my list of things to think about; my offhand reaction is that the class of things that "commute with T" is a lot less than the class of definable clubs. I conjecture that you will not be able to prove that Sigma_3 definable properties "commute with T" for example. I have been working with a class of properties that *do* commute with T which can be characterized as follows. In deciding P(alpha) one is allowed to ask questions about alpha in L_beta where beta is the $n^{th}$ cardinal in L after alpha. This is not the most general class of properties that can be shown to commute with T, but I don't know a clean definition of a maximal class. Module this observation, your sketch looks right to me, but I have **not** thought it through carefully. The whole problem of truly getting Mahlo cardinals in L is that the classes one encounters provably do not commute with T. [Like all offhand remarks take this last with a grain of salt; there is always the possibility that by reconceptualizing this difficulty can be made to go away.] It's hard to characterize the approach I am trying to push through now in a few words. But roughly, it tries to meet the difficulty head on rather than sidestep it. [If it works and doesn't simplify itself it will be a *long* and complicated proof. Not that I don't prefer simple proofs when I can find them!] As ever, Bob From holmes@catseye.idbsu.edu Tue May 2 13:07 MDT 1995 Received: by catseye.idbsu.edu (1.38.193.4/16.2) id AA20448; Tue, 2 May 1995 13:07:06 -0600 Date: Tue, 2 May 1995 13:07:06 -0600 From: Randall Holmes Return-Path: To: holmes@catseye.idbsu.edu, solovay@math.berkeley.edu Subject: Re: My sketch Status: RO I'm well aware that the class of things which commute with T is difficult to capture! This is why the argument I outlined doesn't seem to give any hint of how to get an actual Mahlo cardinal. This week is the last week of the semester here; I'm giving a final exam on Monday and then I will be free to work on writing up the Con(NFUA) -> Con(ZFC) argument. I think that this argument may contain some ideas which are useful for thinking about large cardinals in NFUA; it uses a kind of reflection property of the isomorphism classes of well-founded extensional relations in NFUA, and one might hope that better understanding of reflection properties (there might be better ones in L) might help in looking for things like Mahlo cardinals. --Randall From solovay@math.berkeley.edu Thu May 4 23:49 MDT 1995 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA24285; Thu, 4 May 1995 23:49:22 -0600 Return-Path: Received: by math.berkeley.edu (8.6.10/1.33(math)Ow.1) id WAA17105; Thu, 4 May 1995 22:44:40 -0700 Date: Thu, 4 May 1995 22:44:40 -0700 From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199505050544.WAA17105@math.berkeley.edu> To: holmes@catseye.idbsu.edu In-Reply-To: Randall Holmes's message of Thu, 4 May 1995 10:40:32 -0600 <199505041635.JAA27250@math.berkeley.edu> Subject: But it is involved! Status: RO Randall, I thought some about how to get Con(NFUA) directly from Schmerl's partition property, and it's not clear to me. [I could go from the partition property back to the n Mahlo's and then give my original proof, but that clearly is cheating. Don't worry about this till Monday. I would like to see your proof then if I haven't figured it out by then. As ever, Bob From solovay@math.berkeley.edu Thu May 11 18:44 MDT 1995 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA01857; Thu, 11 May 1995 18:44:21 -0600 Return-Path: Received: by math.berkeley.edu (8.6.10/1.33(math)Ow.1) id RAA09366; Thu, 11 May 1995 17:39:34 -0700 Date: Thu, 11 May 1995 17:39:34 -0700 From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199505120039.RAA09366@math.berkeley.edu> To: holmes@catseye.idbsu.edu Subject: Yes and No Status: RO Randall, Yes, I now see how to get the consistency of NFUA via the Schmerl partition principle. No, I do not quite believe your proof [though obviously, in view of my prior "Yes", I see a repair]. You assert without proof that if f(a_1,...,a_n) = f(a_2,..., a_{n+1}), then it follows that f(a_1,...,a_n) < a_0. I can cook up systems of functions where this is not the case. As ever, Bob From holmes@catseye.idbsu.edu Fri May 12 07:54 MDT 1995 Received: by catseye.idbsu.edu (1.38.193.4/16.2) id AA02087; Fri, 12 May 1995 07:54:40 -0600 Date: Fri, 12 May 1995 07:54:40 -0600 From: Randall Holmes Return-Path: To: holmes@catseye.idbsu.edu, solovay@math.berkeley.edu Subject: Re: Yes and No Status: RO Dear Bob, That assertion (that elements fixed under the automorphism are less than a_i's) _is_ true for any element of V_{a_0} (the domain of the model of NFUA); sorry for not making this clear (obviously it was not clear in my mind that I needed to point this out!). I did already know that other, larger objects in the nonstandard model of set theory being constructed may be fixed under the automorphism (I think that this may be inevitable; I would be interested to see a fix that avoids this, but I don't need it for this proof). But any element of _the model of NFUA_ is either of a rank below any a_i (and so fixed under the automorphism) or is of a rank between a_i's, and so is moved by the automorphism. --Randall From solovay@math.berkeley.edu Fri May 12 10:35 MDT 1995 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA02185; Fri, 12 May 1995 10:35:49 -0600 Return-Path: Received: by math.berkeley.edu (8.6.10/1.33(math)Ow.1) id JAA22967; Fri, 12 May 1995 09:31:02 -0700 Date: Fri, 12 May 1995 09:31:02 -0700 From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199505121631.JAA22967@math.berkeley.edu> To: holmes@catseye.idbsu.edu Cc: holmes@catseye.idbsu.edu In-Reply-To: Randall Holmes's message of Fri, 12 May 1995 07:54:40 -0600 <199505121349.GAA19758@math.berkeley.edu> Subject: Yes and No Status: RO Randall, I'm still not quite convinced. We know that ordinals less than a_0 are fixed. But why does it follow that elements of rank less than a_0 are fixed. I had two repairs, but here is the simplest one. Examining the proof of Schmerl's combinatorial principle shows that we can assume that all the indiscernibles lie in some preassigned stationary subset of the n-Mahlo. Unfortunately, Schmerl doesn't record this fact, but after you have been through the proof of his combinatorial principle, I can explain this refinement. here is how we get the desired sentences: By choosing a suitable club, we can insure the sentences f(a_1,...,a_n) < a_{n+1}. If also f(a_1,...,a_n) = f(a_2,...,a_{n+1}), then it follows that f(a_1,..,a_n) = f(a_{n+2},...,a_{2n+1}). Hence f(a_{n+2},...,a_{2n+1}) < a_{n+1}). So f(a_1,...,a_n) < a_0 as desired. We also can impose that the a_n's are inaccessible cardinals, so that the point I raised in the first paragraph of this letter is also handled. As ever, Bob From holmes@catseye.idbsu.edu Fri May 12 10:53 MDT 1995 Received: by catseye.idbsu.edu (1.38.193.4/16.2) id AA02202; Fri, 12 May 1995 10:53:36 -0600 Date: Fri, 12 May 1995 10:53:36 -0600 From: Randall Holmes Return-Path: To: holmes@catseye.idbsu.edu, solovay@math.berkeley.edu Subject: Re: Yes and No Status: RO Dear Bob, It is definitely necessary to require explicitly that a_i = beth{a_i}, so that the fact about elements of ranks works out correctly! This is enough; the a_i's don't need to be inaccessible (though it is hardly lavish to require this!). At this point you are bringing out things which I "know" implicitly about the construction but am forgetting to state :-( --Randall From solovay@math.berkeley.edu Fri May 12 13:24 MDT 1995 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA02308; Fri, 12 May 1995 13:24:08 -0600 Return-Path: Received: by math.berkeley.edu (8.6.10/1.33(math)Ow.1) id MAA29107; Fri, 12 May 1995 12:19:20 -0700 Date: Fri, 12 May 1995 12:19:20 -0700 From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199505121919.MAA29107@math.berkeley.edu> To: holmes@catseye.idbsu.edu Subject: Re: Yes and No Cc: solovay@math.berkeley.edu Status: RO How do you insure from the statement of the Schmerl principle that a_i = Beth_{a_i}? I don't off hand see how to do this. --Bob From solovay@math.berkeley.edu Fri May 12 13:26 MDT 1995 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA02345; Fri, 12 May 1995 13:26:11 -0600 Return-Path: Received: by math.berkeley.edu (8.6.10/1.33(math)Ow.1) id MAA29156; Fri, 12 May 1995 12:21:22 -0700 Date: Fri, 12 May 1995 12:21:22 -0700 From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199505121921.MAA29156@math.berkeley.edu> To: holmes@catseye.idbsu.edu Subject: Re: Corrections Cc: solovay@math.berkeley.edu Status: RO Randall, I don't have time to look at the latest version of your argument at this instant. I'll try to look at it shortly and comment on it. --Bob From solovay@math.berkeley.edu Sat May 13 13:34 MDT 1995 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA02948; Sat, 13 May 1995 13:34:46 -0600 Return-Path: Received: by math.berkeley.edu (8.6.10/1.33(math)Ow.1) id MAA24191; Sat, 13 May 1995 12:29:57 -0700 Date: Sat, 13 May 1995 12:29:57 -0700 From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199505131929.MAA24191@math.berkeley.edu> To: holmes@catseye.idbsu.edu Cc: holmes@catseye.idbsu.edu In-Reply-To: Randall Holmes's message of Fri, 12 May 1995 13:38:05 -0600 <199505121933.MAA29623@math.berkeley.edu> Subject: Corrections Status: RO Dear Randall, You write: The partition determined by the set of sentences Sigma is set up to categorize a sequence of ordinals (x_1...x_n) not by plugging the x_i's into the formulas but by plugging in the x_i'th fixed point of the beth operator in each case. Thus, the sentences are talking about the fixed points of the beth operator indexed by the ordinals below kappa (which are exactly the fixed points below kappa, of course). The C_alpha's are redefined in the same way. Reply: Yes this does it. I think this way of doing things [via the Schmerl partition relations] is cleaner than my earlier proof via Shelah-Schmerl. [I had to construct a set of indiscernibles in the other approach but got them via "iterated ultraproducts".] I'm still thinking hard about the converse, still optimistic, but still with nothing definitive to report. As ever, Bob From solovay@math.berkeley.edu Mon May 15 23:31 MDT 1995 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA04019; Mon, 15 May 1995 23:31:57 -0600 Return-Path: Received: by math.berkeley.edu (8.6.10/1.33(math)Ow.1) id WAA18648; Mon, 15 May 1995 22:27:02 -0700 Date: Mon, 15 May 1995 22:27:02 -0700 From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199505160527.WAA18648@math.berkeley.edu> To: holmes@catseye.idbsu.edu Subject: finally... Status: RO Randall, Here's a theorem [provable, say, in Peano Arithmetic]: For each integer n, NFUA proves the obvious formalization of "there is a non-Cantorian n-Mahlo cardinal". Joined with my earlier result, this shows that the consistency strength of NFUA is **exactly** ZFC plus the following scheme: For each integer n, there is an axiom: "There is an n-Mahlo cardinal". The proof is somewhat more involved than my earlier result that NFUA proves the existence of inaccessibles in L. Unless I hear a declaration of non-interest, I will start drafting a series of letters communicating the proof. [The metatheory I will work in will be second-order number theory, but I could, if I chose, prove the theorem in the fragment "IDelta_0 + Exp" of Peano Arithmetic.] As ever, Bob From solovay@math.berkeley.edu Mon May 15 23:58 MDT 1995 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA04024; Mon, 15 May 1995 23:58:22 -0600 Return-Path: Received: by math.berkeley.edu (8.6.10/1.33(math)Ow.1) id WAA18993; Mon, 15 May 1995 22:53:32 -0700 Date: Mon, 15 May 1995 22:53:32 -0700 From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199505160553.WAA18993@math.berkeley.edu> To: holmes@catseye.idbsu.edu Subject: Outline of the proof Status: RO Usual disclaimer: I think this proof is right and I've "thought through all the details" in my head, but this is the first time I've committed it "to paper" and a bug may emerge at any point. For the following two definitions, [which I will not give precisely in this outline] the theory in which we work is ZFC + V=L. We will introduce a property of a limit cardinal, lambda, and an increasing n+1-tuple of ordinals xi_0,...,xi_n of the tuple being lambda-good. The intuition is that lambda is a supply of ordinals certified cantorian, and for some j [an elementary embedding] j(xi_i) = xi_{i+1}. [This is by no means the literal definition!] We will be able to prove in NFUA that there are natural ["well-founded, subset absolute"] models of V=L + ZFC + for some limit cardinal lambda, there is an n+1-tuple that is lambda good. [This is a theorem scheme.] By a process reminiscent of the construction of a normal measure on a measurable cardinal, we will be able to show that if there is a good n+1-tuple for lambda, there is a "superb" [definition omitted in this outline] n+1-tuple for lambda. Finally, in ZFC + V=L, we will be able to prove that if lambda is a limit cardinal and xi_0,...,xi_{n} is a superb tuple for lambda, then all the xi_i's are n-Mahlo. [This is a theorem scheme proved by induction on n in the metatheory.] That's the high level outline. Of course, many details must be added to convert it into a proof. As ever, Bob From solovay@math.berkeley.edu Tue May 16 02:09 MDT 1995 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA04037; Tue, 16 May 1995 02:09:04 -0600 Return-Path: Received: by math.berkeley.edu (8.6.10/1.33(math)Ow.1) id BAA21244; Tue, 16 May 1995 01:04:14 -0700 Date: Tue, 16 May 1995 01:04:14 -0700 From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199505160804.BAA21244@math.berkeley.edu> To: holmes@catseye.idbsu.edu Subject: The proof--part 1 Status: RO The goal of this and the next two letters is to define the notions of lambda good and lambda superb. In all these letters, we are working in the theory ZFC + V=L. For this letter, our goal is to introduce the class of local functions. There will be countably many local functions, one for each formula phi of the language of set theory whose free variables are an initial segment of the variables. So let phi be a formula such that the variables occuring free in phi are v_0,...,v_n. We will define a function f_phi:Or^n --> Or. The intuition is that to compute f_phi(alpha_1,...,alpha_n) one goes to L_beta where beta is the least cardinal greater than alpha_1,...,alpha_n and then uses the definition phi. Precisely, if there is exactly one ordinal eta such that L_beta thinks phi(eta,alpha_1,...,alpha_n) then f_phi(alpha_1,...,alpha_n) = eta; otherwise, it equals 0. There is also a notion of a local predicate. This has the normal form f(alpha_1,...,alpha_n) = 0. The crucial property of local functions and predicates is that they are absolute from L_gamma to L whenever gamma is a strong limit cardinal. Moreover, the predicates "is singular" or "is n-Mahlo" are local. Via the usual well-ordering of L, it makes sense to ask if a map from say Or--> L is local, and the following maps are local: The map which assigns to a singular limit ordinal a closed cofinal subset of order type its cofinality. The map which assigns to an inaccessible cardinal which is not n+1-Mahlo a closed cofinal subset which is disjoint from the n-Mahlo cardinals. [These maps should be viewed as 0 outside their natural domains. The local maps are closed under compositions and contain the projection functions. This ends letter 1. From solovay@math.berkeley.edu Tue May 16 10:27 MDT 1995 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA04267; Tue, 16 May 1995 10:27:21 -0600 Return-Path: Received: by math.berkeley.edu (8.6.10/1.33(math)Ow.1) id JAA27817; Tue, 16 May 1995 09:22:30 -0700 Date: Tue, 16 May 1995 09:22:30 -0700 From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199505161622.JAA27817@math.berkeley.edu> To: holmes@catseye.idbsu.edu Subject: The proof--part 2 Status: RO The goal of the present letter is to introduce the concept "lambda good". Here is the context. We are working in the theory ZFC + V=L. lambda is a limit cardinal greater than omega. xi_0,...,xi_n is a tuple of ordinals that is either strictly increasing or strictly decreasing. The tuple is lambda good if the following two conditions are met: (1) whenver (a) f is a local function which maps OR^{n+1} to OR; (b) eta is an ordinal less than lambda; and one of f(eta,xi_0,...,xi_{n-1}) and f(eta,xi_1,...,xi_n) is less than lambda, then they both are and they are equal. (2) whenever (a) f is a local function which maps OR^{n+1} to OR; (b) eta is an ordinal less than lambda; and f(eta,xi_0,...,xi_{n-1}) = f(eta,xi_1,...,xi_n) then the common value is less than lambda. This ends the notion of lambda good. Here are some easy consequences of the definition: 1) Let r be an integer with 1 <= r <= n. Let i_1,..., i_r be integers with 0 <= i_1 < ... < i_r < n. Let k be a positive integer and let j_s = i_s + k for all s with 1 <= s <= r. We suppose also that j_r <= n. [Thus the sequence of j's is just a shift of the sequence of i's.] Let f be a local function with domain OR^{r+1} and eta an ordinal less than lambda. Then the obvious analogues of (1) and (2) above for f(eta,xi_{i_1},...,xi_{i_r}) and f(eta,xi_{j_1},...,xi_{j_r}) obtain. 2) There is a "shift indiscernibility" property for local properties that follows from the way we have defined local properties from local functions. 3) We have lambda <= xi_j whenever 0 <= j <= n. It is easy to see that if kappa is say n+100-Mahlo, then there is a limit cardinal lambda < kappa, and ordinals xi_0,..., xi_n less than kappa such that the xi's are lambda good. We don't need this last observation for our proof, however. So ends part 2 of the proof. From solovay@math.berkeley.edu Tue May 16 10:55 MDT 1995 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA04285; Tue, 16 May 1995 10:55:09 -0600 Return-Path: Received: by math.berkeley.edu (8.6.10/1.33(math)Ow.1) id JAA28799; Tue, 16 May 1995 09:50:18 -0700 Date: Tue, 16 May 1995 09:50:18 -0700 From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199505161650.JAA28799@math.berkeley.edu> To: holmes@catseye.idbsu.edu Subject: The proof--part 3 Status: RO The goal of this letter is to give the definition of a tuple being lambda superb and prove that if there is an n+1-tuple that is lambda good, then there is one that is lambda superb. The context is that of our previous letter. Thus we work in the theory ZFC + V=L. lambda is a strong limit cardinal greater than omega. A tuple xi_0,...,xi_n is lambda superb iff: 1) the tuple is lambda good; 2) whenver eta is an ordinal less than lambda, and f is a local function with domain OR^2 such that f(eta,xi_0) < xi_0, then f(eta,xi_0) < lambda. This completes the definition. Suppose then that xi_0,...xi_n is lambda good. We show how to build a tuple gamma_0,..., gamma_n which is lambda superb. Choose a local function f and an eta < lambda such that (1) f(eta, xi_0) >= lambda; (2) subject to (1), f(eta,xi_0) is as small as possible. Now set gamma_i = f(eta,xi_i). A routine verification [which we leave to the reader] shows that gamma_0,..., gamma_n is a lambda superb n+1-tuple. This ends letter 3. Our next letter will show how to construc models of ZFC + "there is a good n+1-tuple" starting from models of NFUA. From solovay@math.berkeley.edu Tue May 16 12:12 MDT 1995 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA04307; Tue, 16 May 1995 12:12:44 -0600 Return-Path: Received: by math.berkeley.edu (8.6.10/1.33(math)Ow.1) id LAA01640; Tue, 16 May 1995 11:07:49 -0700 Date: Tue, 16 May 1995 11:07:49 -0700 From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199505161807.LAA01640@math.berkeley.edu> To: holmes@catseye.idbsu.edu Subject: The proof--part 4 Status: RO Here we show how to get good tuples starting from models of NFUA. The following is a theorem scheme [in the integer variable n] in NFUA: There is an L inaccessible, Omega, such that L_Omega thinks: There is a limit cardinal lambda, and ordinals xi_0,..., xi_n such that the tuple xi-vector is lambda good. In proving this, I will, as in my earlier argument, adopt the Specker formalism: So Omega is the order-type of the least well-ordering of V. There are analogues of Omega at each level: Omega_n is the analogue of Omega at level n. We may and do assume that Omega_n is an initial segment [proper!] of Omega_{n+1}, and that L_{Omega_n} is a transitive set in L_{Omega_{n+1}. THere is an automorphism j which carries L_{Omega_n} isomorphically onto L_{Omega_{n+1}. Care is needed in working with j, since it is not allowed to directly appear in comprehension axioms. The ordinal Omega need not be inaccessible. However, my earlier proof showed that there is a non-Cantorian ordinal theta < Omega such that theta **is** inaccessible in L. We let theta_i be the analogue of theta at level i. One annoying fact is that we cannot prove that j(theta) > theta. If in fact j(theta) < theta, we reinitialize the notaion, replacing theta_i by theta_{-i} and j by j^{-1}. So we have seen how to insure that [after reinitializing the notation] the Omega_i's are all inaccessible in L. Here ends the current letter. Of course, we are in the midst of the proof of the stated theorem scheme in NFUA. From solovay@math.berkeley.edu Wed May 17 23:08 MDT 1995 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA05186; Wed, 17 May 1995 23:08:48 -0600 Return-Path: Received: by math.berkeley.edu (8.6.10/1.33(math)Ow.1) id WAA01024; Wed, 17 May 1995 22:03:56 -0700 Date: Wed, 17 May 1995 22:03:56 -0700 From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199505180503.WAA01024@math.berkeley.edu> To: holmes@catseye.idbsu.edu Subject: The proof-part 6 Status: RO One minor point. Note that an ordinal alpha is fixed by j iff it is fixed by j^{-1}. So even after reinitializing notation, it is true that if alpha is fixed by j and beta is less than alpha, then beta is fixed by j. We will take as our model of ZFC + V= L the model M = L_{Omega_{n+1}}. The tuple which will eventually be shown to be lambda good for a suitable lambda will be Omega_0,...,Omega_n. The lambda which will actullay be shown to work eventually will be built in M by a certain construction which we now describe. We will define a certain strictly increasing sequence of ordinals . lambda will be the sup of this sequence of ordinals. To start things off, we set lambda_0 = omega. Suppose that lambda_n has been defined. We show how to define lambda_{n+1}. This will be the sup of various ordinals: (a) The least cardinal greater than lambda_i. [This clause will insure that lambda is indeed a limit cardinal and that the sequence of lambda_i's is strictly increasing.] (b) all ordinals of the form f(eta,Omega_0,...,Omega_{n-1}) where f is a local function of the appropriate number of variables, and eta is an ordinal less than lambda_i **and**: f(eta,Omega_0,...,Omega_{n-1}) = f(eta,Omega_1,...,Omega_n). Our construction makes trivial one of the two clauses for lambda-good. Towards the other, let s be the function with domain omega such that s(i) = lambda_i. Of course, s lies in M, since the whole construction has taken place in M. Let s' = j(s). Claim: s = s'. Suppose not toward an eventual contradiction. Clearly, s' is also an omega sequence of ordinals, and since, by the Cantorian axiom, j fixes all the non-negative integers, s'(n) = j(s(n)). So it suffices to prove that all the s(n)'s are fixed by j. Suppose not. Note that the set of integers m of the model M such that s(m) is unequal to s'(m) is a set of M. So if it is non-empty, it has a least member. This least member can't be 0 since s(0) = omega which is fixed by j. So it has the form k+1. Thus to prove our claim that s = s' it suffices to show the following: if lambda_k is fixed by j, then so is lambda_{k+1}. This a good place to end the current letter. From solovay@math.berkeley.edu Wed May 17 23:39 MDT 1995 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA05191; Wed, 17 May 1995 23:39:38 -0600 Return-Path: Received: by math.berkeley.edu (8.6.10/1.33(math)Ow.1) id WAA01532; Wed, 17 May 1995 22:34:46 -0700 Date: Wed, 17 May 1995 22:34:46 -0700 From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199505180534.WAA01532@math.berkeley.edu> To: holmes@catseye.idbsu.edu Subject: The proof--part 7 Status: RO Recall that lambda_k is fixed by j and that we are trying to show that lambda_{k+1} is also fixed by j. Now lambda_{k+1} is the sup of two things (a) the least cardinal greater than lambda_k and a set of ordinals which we will recall in a moment. It is obvious that the cardinal successor of lambda_k is fixed by j, so we turn our attention to the other term in the sup. Godel number the local functions of n+1 variables in some reasonable way. Let D be the set of all pairs such that i is the Godel number of a local function of n+1 variables, and eta is less than lambda_k. It is evident that j(D)= D and that D is pointwise fixed by j. Remark:It is a consequence of V=L and the Cantorian axiom that each set fixed by j is also pointwise fixed by j. But the current special case is particularly clear. Define a function F with domain D as follows. [We are working in the model M.] Let be an element of D. Let f be the local function with Godel number i. Then F() = f(eta,Omega_0,...,Omega_{n-1}). Note that the local function f is absolute to L_{Omega_n}, and hence the function F just described sits in L_{Omega_n}. Let F' = j(F). Clearly F' is a function with domain D. Moreover, using the absoluteness of local functions which we have just recalled, we see that F'() = f(eta,Omega_1,...,Omega_n). Working in M, let E be the subset of D consisting of those points of D at which F and F' take the same value. Then since E is a subset of D, j(E) = E. Let G be the restriction of F to E and let G' be the restriction of F' to E. Then clearly G = G' since they are functions with the same domain which take the same value at every point of their common domain. Moreover, it is evident that G' = j(G), since G' is the restriction of j(F) to j(E). The upshot is that j(G) = G. It then follows that the supremum of the range of G is fixed by j. But this is precisely the other term contributing to the definition of lambda_{k+1}. It is now evident that lambda_{k+1} is fixed by j. Our proof of our claim that s = s' is complete. An immediate corrolary of the claim just proved is that every ordinal less than lambda is fixed by j. Now suppose that f(eta,Omega_0,...,Omega_{n-1}) = theta, where f is a local function and theta and eta are both less than lambda [and hence fixed by j]. We have already seen that j commutes with local functions. Hence applying j to this equality, we get: f(eta, Omega_1,...,Omega_n) = theta. Similarly, if f(eta,Omega_1,...,Omega_n) = theta, [with f, theta, eta as above] then applying j^{-1} we conclude that f(eta,Omega_0,...,Omega_{n-1}) = theta. Our proof that Omega,0,...,Omega_n is lambda good in M is complete. So ends letter 7. From solovay@math.berkeley.edu Thu May 18 01:51 MDT 1995 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA05207; Thu, 18 May 1995 01:51:48 -0600 Return-Path: Received: by math.berkeley.edu (8.6.10/1.33(math)Ow.1) id AAA03152; Thu, 18 May 1995 00:46:35 -0700 Date: Thu, 18 May 1995 00:46:35 -0700 From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199505180746.AAA03152@math.berkeley.edu> To: holmes@catseye.idbsu.edu Subject: The proof--part 8 Status: RO Let me make explicit the definition of some terminology I've been using. A cardinal is 0-Mahlo iff it is inaccessible. A cardinal kappa is n+1-Mahlo iff it is inaccessible and the set of n-Mahlo's less than kappa is stationary in kappa. The following will be shown to be a theorem of ZFC + V=L: Let lambda be a limit cardinal. Let kappa_0,...kappa_{n+1} be lambda superb. Then each of the kappa_i's is n-Mahlo. Once we establish this result, the stated lower bound on the consistency strength of NFUA will be completely proved. Note that it is evident that either all the kappa_i's are n-Mahlo or none of them are. [This uses only that the kappa_i's are lambda good.] The proof is by induction on n and the current letter will consider only the case when n = 0. Without loss of generality, we assume that kappa_0 < kappa_1. We consider various cases [each of which will be shown to be absurd]: Case 1: kappa_0 <= omega. This case can't arise since omega < lambda <= kappa_0 Case 2: kappa_0 is not a cardinal. There is a local function f such that f(0,gamma) is the cardinal of gamma. So f(0,kappa_0) < kappa_0. Since the kappa's are lambda superb, we have f(0,kappa_0) < lambda. But lambda is a limit cardinal. So it follows that kappa_0 < lambda which is absurd. Case 3: kappa_0 is a sucessor cardinal. There is a local function f such that if kappa is a successor cardinal, then f(0,kappa) is the cardinal predecessor of kappa. So f(0, kappa_0) < kappa_0. Since the kappa's are lambda superb, f(0,kappa_0) < lambda. But lambda is a limit cardinal. It follows that kappa_0 < lambda which is absurd. Case 4: kappa_0 is a singular strong limit cardinal. There is a local function f such that f(0,kappa) = cf(kappa). Since the kappa's are lambda superb, it follows that cf(kappa_0) = cf(kappa_1) = gamma for some gamma < lambda. There is a local function g which does the following: Let kappa be singular and let eta < cf(kappa). Let h be the L-least map of cf(kappa) cofinally into kappa. Then g(eta,kappa) = h(eta). Now let h_0 and h_1 be the L-least maps of gamma cofinally into kappa. Since the kappa's are lambda superb, we have h_0(eta) = g(eta,kappa_0) = g(eta,kappa_1) = h_1(eta). The upshot is that h_0 = h_1. But then kappa_0 = kappa_1 which is absurd. Since each of these cases has led to a contradiction, we conclude that kappa_0 and kappa_1 are inaccessible cardinals. This completes the basis part of our inductive proof and with it letter 8. From solovay@math.berkeley.edu Thu May 18 02:27 MDT 1995 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA05216; Thu, 18 May 1995 02:27:32 -0600 Return-Path: Received: by math.berkeley.edu (8.6.10/1.33(math)Ow.1) id BAA03703; Thu, 18 May 1995 01:22:40 -0700 Date: Thu, 18 May 1995 01:22:40 -0700 From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199505180822.BAA03703@math.berkeley.edu> To: holmes@catseye.idbsu.edu Subject: The proof--part 9 Status: RO The situation is the following. We are working in the theory ZFC + V=L. We know that whenever lambda is a strong limit cardinal, and gamma_0,...,gamma_{n+1} is lambda superb, then the gamma_i's are n-Mahlo. We are given kappa_0,...kappa_{n+2} which are lambda superb. We have to show that they are all n+1-Mahlo. It is evident that kappa_0,..., kappa_{n+1} are lambda superb. Hence we know [using our inductive hypothesis] that all the kappa_i's are n-Mahlo. Suppose towards a contradiction that not all the kappa_i's are n+1-Mahlo. It follows that none of the kappa_i's are n+1-Mahlo. What does this mean. it means that for each i, there is a set C_i which is club in kappa_i and which contains no n-Mahlo cardinal. We take C_i to be, in fact, the L-least subset of kappa_i with this property. Without loss of generality, we assume that kappa_0 < kappa_1 < ... < kappa_{n+2}. Note that C_0 is unequal to C_1 intersected with kappa_0. For if not, C_1 would have kappa_0 as a limit point and hence kappa_0 would be a member of C_1. But kappa_0 is n-Mahlo, and C_1 contains no n-Mahlo cardinals, which gives a contradiction. There is a local function f:OR^3 --> OR with the following property: If kappa_0 and kappa_1 are not n+1-Mahlo but are n-Mahlo, and C_0 and C_1 are the L-least witnesses that kappa_0 and kappa_1 are not n+1-Mahlo, then f(0,kappa_0,kappa_1) is the least gamma which is in precisely one of C_0 and C_1. Our preceding discussion has shown that f(0,kappa_0,kappa_1) < kappa_0. We show next that f(0,kappa_0,kappa_1) is >= lambda. Suppose not. Then f(0,kappa_0,kappa_1) = f(0,kappa_1,kappa_2). We consider two cases. Case 1: f(0,kappa_0,kappa_1) is in C_0. Then by shift indiscernibility, f(0, kappa_1,kappa_2) is in C_1. But f(0,kappa_0,kappa_1) = f(0,kappa_1,kappa_2). So f(0,kappa_0,kappa_1) is in C_1. But the ordinal f(0,kappa_0,kappa_1) is in precisely one of C_0 and C_1. Contradiction. Case 2: f(0,kappa_0,kappa_1) is in C_1. This case can be handled exactly like case 1. Now define a local function g and an ordinal eta by the following requirements: g(eta,kappa_0,kappa_1) is >= lambda, and is as small as possible subject to that. eta is as small as possible subject to this constraint, and modulo the choice of eta, g has minimal Godel number. Let gamma_i = g(eta,kappa_i, kappa_{i+1}). Then it is easy to see that gamma_0,..., gamma_{n+1} are lambda superb. Hence by induction hypothesis, they are n-Mahlo. Notice that g(eta,kappa_0,kappa_1) <= f(0,kappa_0,kappa_1). It follows that gamma_0 < kappa_0. Since gamma_0 is n-Mahlo, it is not in C_0. Since C_0 is closed, we define an ordinal theta_0 as the sup of C_0 intersect gamma_0. Since theta_0 < gamma_0 and has the form h(eta', kappa_0, kappa_1) for some local h and some eta' < lambda, we have theta_0 < lambda. Now consider theta* which is the least member of C_0 greater than theta_0. Then theta* can be obtained from theta_0 and kappa_0 by applying a local function. Since the kappa_i's are lambda superb, and theta_0 is less than lambda, we conclude that theta* is less than lambda. But clearly, gamma_0 < theta*. So gamma_0 is less than lambda, which is contrary to the definition of the gamma's. Our assumption that the kappa's are not n+1-Mahlo has led to a contradiction. This completes the proof and with it letter 9. From solovay@math.berkeley.edu Thu May 18 02:57 MDT 1995 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA05221; Thu, 18 May 1995 02:57:01 -0600 Return-Path: Received: by math.berkeley.edu (8.6.10/1.33(math)Ow.1) id BAA03753; Thu, 18 May 1995 01:27:05 -0700 Date: Thu, 18 May 1995 01:27:05 -0700 From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199505180827.BAA03753@math.berkeley.edu> To: holmes@catseye.idbsu.edu Subject: All done Status: RO I've sent you nine letters which contain my proof of the optimal lower bound on the cosistency strength of NFUA. If you have any questions, I'll be glad to try to answer them. I will be off the net from this coming Sunday [at some point] till Monday May 29th. The next step should be to prove that NFUB is much stronger in consistency strength than a weakly compact. [This is a conjecture at this point.] But I am going to abandon NFish stuff for a while and return to the pursuit of competence in chess and in physics. As ever, Bob From solovay@math.berkeley.edu Wed May 17 14:31 MDT 1995 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA04761; Wed, 17 May 1995 14:31:45 -0600 Return-Path: Received: by math.berkeley.edu (8.6.10/1.33(math)Ow.1) id NAA06758; Wed, 17 May 1995 13:26:05 -0700 Date: Wed, 17 May 1995 13:26:05 -0700 From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199505172026.NAA06758@math.berkeley.edu> To: holmes@catseye.idbsu.edu Subject: Some comments on the proof letter 5] Status: RO The comments are really rather minor: 1) Though I think it's correct that the proof can be formalized in IDelta_0 + Exp, I wish to retreat from officially claiming it. [There are some unplesant details to think through.] I do still claim that the proof can be formalized in Peano arithmetic and indeed in Primitive Recursive Arithmetic, though officially, I am still presenting a proof in second order number theory. 2) I said that the inductive claim that if xi_0,...,xi_n are lambda superb then the xi_i's are n-Mahlo was a theorem scheme. But it is easy to prove this as an outright theorem in ZFC + V=L. There has to be a scheme somewhere since NFUA does not prove "For all integers n, ZFC + there exists an n-Mahlo is consistent". [By Godel.] But the scheme is the result whose proof was started in letter 4. 3) I envisage four more letters to the proof. The proof started in letter 4 will take two more letters. The proof that if an n+1-tuple is lambda superb its components are n-Mahlo will take two letters. I hope that these letters are getting through! As ever, Bob From solovay@math.berkeley.edu Tue May 30 13:43 MDT 1995 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA02097; Tue, 30 May 1995 13:43:48 -0600 Return-Path: Received: by math.berkeley.edu (8.6.10/1.33(math)Ow.1) id MAA11789; Tue, 30 May 1995 12:38:23 -0700 Date: Tue, 30 May 1995 12:38:23 -0700 From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199505301938.MAA11789@math.berkeley.edu> To: holmes@catseye.idbsu.edu Subject: Did you believe it? Status: RO Here "it" refers to my proof from NFUA that there are n-Mahlo cardinals in L? I did think though the issue of carrying out the proof in a weak meta-theory and am now prepared to assert that the proof can be carried out in IDelta_0 + Exp. I definitely can't do the proof in any substantially weaker metatheory. [I make no claim that the issue of what the weakest metatheory in which this half of the equiconsistency can be proved has any significance.] While writing this, I realize that it is not evident that the other half of the equiconsistency doesn't need a stronger metatheory. I will ponder this [admittedly esoteric] issue offline. As ever, Bob From solovay@math.berkeley.edu Tue May 30 17:58 MDT 1995 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA02189; Tue, 30 May 1995 17:58:29 -0600 Return-Path: Received: by math.berkeley.edu (8.6.10/1.33(math)Ow.1) id QAA18518; Tue, 30 May 1995 16:53:03 -0700 Date: Tue, 30 May 1995 16:53:03 -0700 From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199505302353.QAA18518@math.berkeley.edu> To: holmes@catseye.idbsu.edu Cc: holmes@catseye.idbsu.edu In-Reply-To: Randall Holmes's message of Tue, 30 May 1995 14:30:39 -0600 <199505302025.NAA13327@math.berkeley.edu> Subject: A very minor point Status: RO I looked at the proof of Forall n Con(ZFC + exists an n-Mahlo) --> Con(NFUA); offhand, I don't see how it goes in IDelta_0 + Exp; I can do it in PRA and indeed in the theory IDelta_0 + "The stack-of-twos function is total". Here the stack of twos function is the function J such that J(0) = 0 and J(n+1) = 2^{J(n)}. [So J(5) is a tad more than 65,000. Perhaps there are sentences of arithmetic whose proofs in NFU are much shorter than their proofs in type theory. This seems likely to me now, but I certainly haven't thought through a proof of this. [The analogous statement is true for GB and ZF.] \end{verbatim} \end{document}From aki@math.bu.edu Fri Feb 16 14:06 MST 1996 Received: from MATHSRV.BU.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA01654; Fri, 16 Feb 1996 14:05:54 -0700 Return-Path: Received: from math.bu.edu (math.bu.edu [128.197.60.50]) by mathsrv.bu.edu (8.7.1/8.7.1) with ESMTP id QAA02001 for ; Fri, 16 Feb 1996 16:05:09 -0500 (EST) From: Akihiro Kanamori Received: (from aki@localhost) by math.bu.edu (8.7.1/8.7.1) id QAA00490; Fri, 16 Feb 1996 16:05:23 -0500 (EST) Date: Fri, 16 Feb 1996 16:05:23 -0500 (EST) Message-Id: <199602162105.QAA00490@math.bu.edu> To: holmes@catseye.idbsu.edu In-Reply-To: <199602161854.NAA17247@cs.bu.edu> (message from Randall Holmes on Fri, 16 Feb 1996 11:55:56 -0700) Subject: Re: Subtle cardinals Status: R Dear Randall, I do not offhand know the answer to your question. Depending on what iteration of my book volume II you might have, I do show that the regressive function characterization of subtle cardinals does have that sort of robustness: Without any reference to club sets, just a regressive partition relation requiring finite homogeneous sets requires subtlety. See Theorem 37.7 of my book, or my paper in APAL 52 (1991), 65--77. The regressive partition characterization of subtlety, 36.14, alas, depends on the club sets in the subtlety definition. The first order of business would be to get set coherence as in 36.12 not only for two ordinals \alpha and \beta, but many. --Aki Kanamori p.s. I thought you do mainly NF and computer science? From solovay@math.berkeley.edu Fri Feb 23 14:31 MST 1996 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA10254; Fri, 23 Feb 1996 14:31:22 -0700 Return-Path: Received: by math.berkeley.edu (8.7.1/1.33(math)Ow.2) id NAA00646; Fri, 23 Feb 1996 13:31:07 -0800 (PST) Date: Fri, 23 Feb 1996 13:31:07 -0800 (PST) From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199602232131.NAA00646@math.berkeley.edu> To: holmes@catseye.idbsu.edu In-Reply-To: Randall Holmes's message of Fri, 23 Feb 1996 11:47:29 -0700 <199602231847.KAA25746@math.berkeley.edu> Subject: NFUB Status: RO Randall, My intuition re NFUB is not that sharp. But it's the kind of result I would expect: Versions of NF should not be equiconsistent with ZFC + a single large cardinal but with a scheme [like my result for NFUA]. This incidentally was why I didn't believe your result about NFUA being equiconsistent with ZFC when you told it to me in Boise. Alas, I am utterly frantic and will remain so till around July or August [when we move into our projected new house in Eugene] so I am unlikely to read your proof when I first get it. But I would like to get it. Your new result nicely complements my result about NFUA. --Bob From aki@math.bu.edu Fri Feb 23 14:49 MST 1996 Received: from MATHSRV.BU.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA10280; Fri, 23 Feb 1996 14:49:47 -0700 Return-Path: Received: from math.bu.edu (math.bu.edu [128.197.60.50]) by mathsrv.bu.edu (8.7.1/8.7.1) with ESMTP id QAA07064 for ; Fri, 23 Feb 1996 16:49:00 -0500 (EST) From: Akihiro Kanamori Received: (from aki@localhost) by math.bu.edu (8.7.1/8.7.1) id QAA11299; Fri, 23 Feb 1996 16:49:12 -0500 (EST) Date: Fri, 23 Feb 1996 16:49:12 -0500 (EST) Message-Id: <199602232149.QAA11299@math.bu.edu> To: holmes@catseye.idbsu.edu In-Reply-To: <199602231916.OAA04562@math.bu.edu> (message from Randall Holmes on Fri, 23 Feb 1996 12:16:56 -0700) Subject: Re: Your help Status: RO I hope that it works out. In my 1991 paper, I primarily focus on the n-subtle and n-Mahlo cardinals in connection with Borel diagonalization. Perhaps there is a direction connection between that and NFU? --Aki From tomek@diamond.idbsu.edu Wed Mar 6 13:45 MST 1996 Received: from diamond.idbsu.edu by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA26022; Wed, 6 Mar 1996 13:45:02 -0700 Return-Path: Received: by diamond.idbsu.edu (1.37.109.16/16.2) id AA280955179; Wed, 6 Mar 1996 13:46:19 -0700 Date: Wed, 6 Mar 1996 13:46:19 -0700 From: Tomek Bartoszynski To: holmes@diamond.idbsu.edu Subject: paper Status: RO Here is the paper that I mentioned and the theorem that I had in mind --tomek @article{ HarShel85Som , number ={2} , journal ={Notre Dame Journal of Formal Logic} , pages ={178--188} , volume ={26} , title ={Some exact equiconsistency results in set theory} , year ={1985} , author ={Harrington, Leo and Shelah, Saharon} } \begin{lemma}[\cite{HarShel85Som}] Suppose that $\kappa$ is a regular cardinal which is not a weakly compact cardinal in ${\mathbf L}$. Then, in ${\mathbf L}$, there is a tree $T$ on $\kappa$ such that for any $M \models \ZFCa$, if $M \models \hbox{``}\,T$ has a branch of length $\kappa$'' then $M \models \cf(\kappa) = \boldsymbol\aleph_{0}$. \end{lemma} \Proof Work in ${\mathbf L}$. Since $\kappa$ is not weakly compact in ${\mathbf L}$ there is a $\kappa$-tree $T_{0}$ which does not have $\kappa$ branches. In other words, $T_{0}$ is a $\kappa$-Aronszajn tree. Without loss of generality, we can assume that elements of the $\alpha$-th level of $T_{0}$ are functions from $\alpha$ to $\alpha$. Define a tree $T$ as follows: $\eta = \langle \alpha, M,b \rangle \in T$ if \begin{enumerate} \item $\alpha < \kappa , \ b$ is a function , $\dom(b) \supseteq \alpha ,\ b \rest \alpha \in T_{0}$, \item $M = {\mathbf L}_{\beta}$ for some $\beta,\ \alpha \subseteq M, \ b \in M$, and \item $M=$ Skolem hull of $\alpha \cup \left\{b\right\}$. \end{enumerate} If $\eta = \langle \alpha, M,b \rangle$ and $\tau = \langle \alpha', M',b' \rangle$ are elements of $T$, then $\tau \geq \eta$ if $\alpha' \geq \alpha$ and $M$ is equal to the transitive collapse of the Skolem hull of $\alpha \cup \left\{b'\right\}$ in $M'$ and $b'$ collapses to $b$. Clearly, $T$ is a tree and the element $\langle \alpha, M,b \rangle$ is on level $\alpha$. If $\left\{\langle \alpha, M_{\alpha},b_{\alpha} \rangle : \alpha < \kappa\right\} $ is a branch through $T$, then by identifying $M_{\alpha}$ with the Skolem hull of $\alpha \cup \left\{b_{\beta}\right\}$ inside $M_{\beta}$ for $\beta \geq \alpha$, we obtain an elementary chain of structures $\left\{\langle M_{\alpha}, b_{\alpha}, \in \rangle : \alpha < \kappa\right\}$. Let $\langle M, b, E \rangle$ be the direct limit of this sequence. Clearly $$\langle M, b, E \rangle \models {\mathbf V} = {\mathbf L} \ \& \ b \hbox{ is a function } \& \ \kappa \subseteq M \ \& \ \forall \alpha<\kappa \ b \rest \alpha \in T_{0} .$$ Therefore, $b \rest \kappa$ is a branch through $T_{0}$. Thus, $b \not \in {\mathbf L}$ and we get that $M$ is not well-founded. But $M$ is a direct limit of well-founded structures so this limit must have length of cofinality $\omega$. Thus, $\kappa$ has cofinality $\omega$. $\QED$ From solovay@math.berkeley.edu Thu Jul 11 17:16 MDT 1996 Received: from math.berkeley.edu by diamond.idbsu.edu with ESMTP (1.37.109.16/16.2) id AA147456996; Thu, 11 Jul 1996 17:16:36 -0600 Return-Path: Received: by math.berkeley.edu (8.7.5/1.33(math)Ow.2) id QAA24671; Thu, 11 Jul 1996 16:14:37 -0700 (PDT) Date: Thu, 11 Jul 1996 16:14:37 -0700 (PDT) From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199607112314.QAA24671@math.berkeley.edu> To: holmes@diamond.idbsu.edu Subject: Re: NFUB etc. Cc: solovay@math.berkeley.edu Status: RO Randall, The following should be true, but unlike the other result, I have *never* checked the details. One formulation of completely ineffable is in terms of player II winning a certain closed game. Let G(n) be the proposition that player II can play n moves without losing immediately. Then I think I see how to prove that if ZFC + G(n) is consistent for all n [view this statement as a scheme!] then NFUB is consistent. Roughly speaking we would now construct the ultrafilter we use in an ultrapower of V rather than a generic extension. [I would probably work out the formal details slightly differently.] I suspect that the statements G(n) form a strict hierarchy in consistency strength and that it is much stronger than the hierarchy of n-ineffables, but at the moment, I don't have a clue as to how to prove this last claim. I *am* optimistic that I can figure out what's going on. --Bob From solovay@math.berkeley.edu Fri Jul 12 09:47 MDT 1996 Received: from math.berkeley.edu by diamond.idbsu.edu with ESMTP (1.37.109.16/16.2) id AA221176459; Fri, 12 Jul 1996 09:47:39 -0600 Return-Path: Received: by math.berkeley.edu (8.7.5/1.33(math)Ow.2) id IAA09806; Fri, 12 Jul 1996 08:45:40 -0700 (PDT) Date: Fri, 12 Jul 1996 08:45:40 -0700 (PDT) From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199607121545.IAA09806@math.berkeley.edu> To: holmes@diamond.idbsu.edu Subject: Re: NFUB etc. Cc: solovay@math.berkeley.edu Status: RO Randall, The throes of moving have cut me off from all libraries. Isn't it in Kanamori's paper? What definition for completely ineffable does he give? I don't recall if I spelled out the game-theoretic definition in my prior letter. Here it is. There are two players who alternate moves. There are restraints on player II's moves, so he could well find himself unable to make a legal move. If so, he loses immediately. If he can play the whole game without "losing immediately" II wins. So the game is open for I, and hence determined. If player II wins, the cardinal kappa is said to be "completely ineffable". [In the preceding sentence, "II wins" means "II has a winning strategy".] I has two sorts of moves. In the first, he plays a kappa sequence of su -sets of kappa; in the second he plays a function from kappa to kappa. II has only one type of move. He plays a subset of kappa of order type kappa. II's moves are subject to the following three constraints. 1) The set he plays on a given turn is a subset of the sets he has played on prior turns. 2) If I plays a kappa sequence of sets, then a tail of the set [begin after the then!] for each subset in this array, a tail of the set played by II is either (a) in this set or (b) out of this set. 3) If I plays a function and the function is regressive on a tail of the set played by II, then it is constant on a tail of the set played by II. the following example may be instructive. Suppose kappa is a measurable cardinal. Let mu be a normal measure on kappa. II can win this game by playing suitable sets of mu measure 1. [Exercise!] As ever, --Bob From solovay@math.berkeley.edu Fri Jul 12 20:53 MDT 1996 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA02336; Fri, 12 Jul 1996 20:53:30 -0600 Return-Path: Received: by math.berkeley.edu (8.7.5/1.33(math)Ow.2) id TAA26055; Fri, 12 Jul 1996 19:49:33 -0700 (PDT) Date: Fri, 12 Jul 1996 19:49:33 -0700 (PDT) Message-Id: <199607130249.TAA26055@math.berkeley.edu> From: "Robert M. Solovay" To: holmes@catseye.idbsu.edu Subject: Preannouncement Status: RO I am close to proving the following conjecture of yours. Assume that for all n the theory ZFC + "there exists an n-ineffable cardinal" is consistent; then NFUB is consistent. Here is the part that I am sure about at the moment. Let n be a positive integer. Then there is a positive integer m [4n + 4 should work] so that if kappa is m-ineffable, then there is an inaccessible alpha < kappa such that alpha satisfies G(n). [My weakening of completely ineffable.] The proof is rather like the proof that if kappa is an Erdos cardinal then there is an alpha < kappa which is completely ineffable. [Alas, I figured though I think this result is "well known".} From solovay@math.berkeley.edu Fri Jul 12 21:02 MDT 1996 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA02340; Fri, 12 Jul 1996 21:02:27 -0600 Return-Path: Received: by math.berkeley.edu (8.7.5/1.33(math)Ow.2) id TAA26165; Fri, 12 Jul 1996 19:58:29 -0700 (PDT) Date: Fri, 12 Jul 1996 19:58:29 -0700 (PDT) Message-Id: <199607130258.TAA26165@math.berkeley.edu> From: "Robert M. Solovay" To: solovay@math.berkeley.edu Cc: holmes@catseye.idbsu.edu In-Reply-To: <199607130249.TAA26055@math.berkeley.edu> (solovay@math.berkeley.edu) Subject: Re: Preannouncement Status: RO Randall, my lap top does not interact well with emacs. To finish the previous letter if one joins the result of that letter with my previous suspicion that if for all n, ZFC + there is a cardinal satisfying G(n) is consistent, then so is NFUB, one gets the pre-announced result. I will have to think through the proposed improvement to my result about NFUB from completely ineffables and check that it actually does work. As ever, Bob From solovay@math.berkeley.edu Fri Jul 12 21:35 MDT 1996 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA02345; Fri, 12 Jul 1996 21:35:58 -0600 Return-Path: Received: by math.berkeley.edu (8.7.5/1.33(math)Ow.2) id UAA26704; Fri, 12 Jul 1996 20:32:00 -0700 (PDT) Date: Fri, 12 Jul 1996 20:32:00 -0700 (PDT) Message-Id: <199607130332.UAA26704@math.berkeley.edu> From: "Robert M. Solovay" To: solovay@math.berkeley.edu Cc: holmes@catseye.idbsu.edu In-Reply-To: <199607130258.TAA26165@math.berkeley.edu> (solovay@math.berkeley.edu) Subject: Re: Preannouncement Status: RO Randall, To blow cold on my preannouncement. I am having trouble seeing how to get a model of NFUB from a completely ineffable. If that proof goes down obviously the proposed improvement goes down with it. Alas, I must get back to the dull business of moving so it may be a few days before I know what's going on. --Bob I'm thinking about what can be done inside NFUB itself, except that I'm using an additional assumption: (*) for each alpha in Ord, either alpha is s.c. (strongly Cantorian) or alpha > T^n(Omega) for some natural number n (Omega is the order type of the ordinals, T is the external operation on ordinals induced by the "singleton image" operation on well-orderings). (one could equally well talk about isomorphism classes of well-founded relations here). I don't know whether this strengthens NFUB; it is true in the model of NFUB built with a measurable cardinal, and it is true in the model of NFUA that we described in our correspondence. Definition: call a set A of ordinals "natural" iff x \in A iff Tx \in A. Claim: In NFUB + extra assumption above, each set of ordinals has the same s.c. elements as a uniquely determined natural set. Corollary: Using the main axiom scheme of NFUB, each proper class of ordinals definable by a formula contains the same s.c. elements as a uniquely determined natural set. Proof of Claim: Take any set A. Let T[A] be the image of A under the T operation. If A \cap T[A] = T[A], we are done (A is natural). Otherwise, consider the first element x of the symmetric difference of A and T[A]. x clearly cannot be s.c. x > T^n(Omega) for some n by the assumption above. T^{-n-1}[A] will be the desired natural set. This means that NFUB + * allows us to associate a unique set with each definable class of s.c. ordinals. This means that we can define classes of s.c. ordinals freely using (in effect) quantification over all classes of s.c. ordinals. Call the (external) order type of the s.c. ordinals kappa. We can interpret the theory of sets hereditarily of size kappa in NFUB. (sets of size kappa of subsets of kappa can be represented as subsets of kappa x kappa using the natural pairing relation on the s.c. ordinals). This should provide an interpretation of ZF - power set (it takes work to get choice). Further, we can define an external ultrafilter on subsets of kappa: a class of s.c. ordinals belongs to the ultrafilter iff the associated natural set contains Omega (we could use any other fixed non-Cantorian ordinal). It is straightforward to show that this ultrafilter is nonprincipal and kappa-complete (we can represent a set of lambda Return-Path: To: solovay@math.berkeley.edu Subject: n-ineffables Cc: holmes@diamond.idbsu.edu Status: R Dear Bob, I think I can prove that n-ineffables exist in NFUB; not just _each_ of them but _all_ of them (NFUB |= "there are n-ineffables for each n"); I do this by showing that there are n-subtle cardinals for each n, by showing that regressive functions on the infinite ordinals have large homogeneous sets. The result isn't just for _each_ n, because NFUB has mathematical induction on unstratified conditions. The argument is a refinement of the argument for n-Mahlo cardinals in NFUA outlined in my slides (this part of the LaTeX file for the slides is appended). For example, to show that a regressive function on [Ord]^2 has a homogeneous set of noncantorian size rather than just a min-homogeneous set of noncantorian size, we proceed as follows: for any node in the ramification tree of the function (the function being restricted as described in the slides so that it will commute with T), we define val(x) as the value of the function at any pair of values x 2$, consider the partition of the $(n-1)$-element subsets of the non-Cantorian branch effected by adding any larger $n$th element and applying $f$; then proceed by (meta-)induction. Nothing in this argument depends on anything about the domain of $f$ except that it is non-Cantorian and consists of infinite ordinals. The min-homogeneous set finally obtained will be non-Cantorian, which more than fulfils the need for $n+5$ elements! Notice that the process here is unstratified (the restriction of $f$ at each stage cannot be described uniformly in {\em NFU\/}) and so works only for concrete natural numbers $n$, as one would hope. \end{slide*} \end{document} From solovay@math.berkeley.edu Tue Jul 23 11:37 MDT 1996 Received: from math.berkeley.edu by diamond.idbsu.edu with ESMTP (1.37.109.16/16.2) id AA031883422; Tue, 23 Jul 1996 11:37:02 -0600 Return-Path: Received: by math.berkeley.edu (8.7.5/1.33(math)Ow.2) id KAA06617; Tue, 23 Jul 1996 10:34:35 -0700 (PDT) Date: Tue, 23 Jul 1996 10:34:35 -0700 (PDT) Message-Id: <199607231734.KAA06617@math.berkeley.edu> From: "Robert M. Solovay" To: holmes@diamond.idbsu.edu In-Reply-To: <199607231523.IAA03612@math.berkeley.edu> (message from Randall Holmes on Tue, 23 Jul 1996 09:25:55 -0600) Subject: Re: omega-Erdos Status: RO Randall, I owe you a letter which I hope to get to today, but this is just a quick reply to your latest missive. I no longer claim to get NFUB from either a completely ineffable or from an omega-Erdos. Since you are saying you can get it from the latter, let me sketch the difficulty I see. One will apply the omega Erdos, getting a set of indiscernibles which are good over some inaccessible cardinal less than the omega-Erdos. Call this smaller cardinal alpha, and call the omega Erdos kappa. The indiscernibles will yeild a certain subset of alpha which I will call the blueprint. The blueprint will directly know about certain subsets of alpha but at most alpha of them. The plan would be to use all this to get a model of NFUB where the strongly Cantorian ordinals would be those less than alpha. Now suppose using this model we define a new subset of alpha. I see no reason why it should be one of the subsets of alpha represented in the blueprint. Note that the situation is **different** for bounded subsets of alpha since every such set is coded in our blueprint. I hope this is not too cryptic. Of course, your proof may get around this difficulty in some way that I'm not currently seeing. --Bob From holmes@diamond.idbsu.edu Tue Jul 23 12:31 MDT 1996 Received: by diamond.idbsu.edu (1.37.109.16/16.2) id AA039026715; Tue, 23 Jul 1996 12:31:55 -0600 Date: Tue, 23 Jul 1996 12:31:55 -0600 From: Randall Holmes Return-Path: To: holmes@diamond.idbsu.edu, solovay@math.berkeley.edu Subject: Re: omega-Erdos Status: RO I have a similar problem, which I naturally discovered as soon as I had my moment of "insight" and sent you my brief note. The approach I am taking is to consider building a model of V_kappa with indiscernibles using kappa weakly compact (which gets refined to omega-Erdos later). In this model, the s.c. ordinals are standard. One can add a set "realizing" any class of s.c. ordinals one wants (for example, one defined by your favorite unstratified formula); the resulting model will have a set with the s.c. elements of this class, satisfy the same stratified formulas, but not necessarily the same unstratified formulas... The omega-Erdos-ness of kappa figures in a device which I was working on to try to keep the unstratified formula stable--which doesn't work so far. --Randall From solovay@math.berkeley.edu Fri Jun 13 00:46 MDT 1997 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA00304; Fri, 13 Jun 1997 00:46:41 -0600 Return-Path: Received: by math.berkeley.edu (8.7.5/1.33(math)Ow.3) id XAA20681; Thu, 12 Jun 1997 23:39:07 -0700 (PDT) Date: Thu, 12 Jun 1997 23:39:07 -0700 (PDT) From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199706130639.XAA20681@math.berkeley.edu> To: holmes@catseye.idbsu.edu Subject: Some terminology Cc: solovay@math.berkeley.edu Status: RO Randall, in the next letter, I will announce some new upper and lower bounds on the consistency strength of NFU. But I first wanted to introduce some terminology and make a remark about my "old results". By NFUB+ I mean the variant introduced in your letter where the ordinals T^n[\Omega] are downward cofinal in the non-Cantorian ordinals. By NFUB- I mean the following extension of NFUA [+ AC if that needs saying]. If alpha is a Cantorian ordinal and phi(x) is any formula [with one free variable, but we allow other free variables to have been plugged by parameters] then the set of members of alpha satisfying phi is a set. It should be evident that [at least if AC is included in NFUB] that NFUB- is weaker than NFUB. [Note that phi is not required to be stratified.] I think my result from a completely ineffable was that NFUB- was consistent. I have not bothered to reconstruct this result, but I think this is the explanation for my former error. I believe I have shown recently the weaker result that ZFC + "There is an Erdos cardinal" proves the consistency of NFUB-; this is a relatively easy result. ---Bob From solovay@math.berkeley.edu Fri Jun 13 01:09 MDT 1997 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA00310; Fri, 13 Jun 1997 01:09:01 -0600 Return-Path: Received: by math.berkeley.edu (8.7.5/1.33(math)Ow.3) id AAA20997; Fri, 13 Jun 1997 00:01:26 -0700 (PDT) Date: Fri, 13 Jun 1997 00:01:26 -0700 (PDT) From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199706130701.AAA20997@math.berkeley.edu> To: holmes@catseye.idbsu.edu Subject: NFUB's consistency strength Cc: solovay@math.berkeley.edu Status: RO All these results have only had a "first reading" in my head. I plan to write the proofs up; I'll let you know if the proofs don't survive closer inspection. Some more terminology. If kappa is a regular cardinal, and alpha is an ordinal less than kappa^+ then the notion "kappa is alpha-Mahlo" is standard and extends the notion of n-Mahlo that I used previously. It is true that if kappa is beta-Mahlo and alpha is less than beta, then kappa is alpha Mahlo. For alpha < kappa, the notion is exactly what one would suspect; for alpha >= kappa, the concept is a hair more subtle. It is true that if kappa is weakly compact, then kappa is alpha Mahlo for all alpha < kappa^+. Essentially, I can prove that if NFUB is consistent, then for "any reasonable definition" of alpha, ZFC + "there is a kappa that is alpha Mahlo" is consistent. As a sample, take alpha to be the least ordinal > kappa such that L_alpha is a model of ZFC [where kappa is understood to be a regular cardinal]. The proof of these lower bounds is not difficult. Significantly harder is the upper bound. The theory ZFC + "There is a cardinal which is simultaneously weakly compact and Erdos" proves the consistency of NFUB. As a corollary, NFUB does not prove that 0# exists. My feeling is that the current lower bound is way too weak and that the upper bound is essentially optimal. Indeed I *conjecture* that the following theory is equiconsistent with NFUB: ZFC- + V=L + "There is a cardinal kappa which is simultaneously weakly compact and Erdos". Call the displayed theory T. Then the upper bound proof easily adapts to show that Con(T) ==> Con(NFUB). The lower bounds factor through a proof that Con(NFUB) ==> Con(T_1) where T_1 is the theory ZFC- + V=L + "There is a weakly compact cardinal". Very roughly, the weakly compact cardinal is the order-type of the class of Cantorian ordinals. What I need to show to get an exact consistency calculation is that this order-type is also an Erdos cardinal. I have some ideas re this but nothing like a proof as yet. If I understand what you are claiming re NFUB+, then you can show that NFUB+ entails 0# exists [and even that ZFC + "There are a proper class of Ramsey cardinals" is consistent.] So NFUB+ would be **much** stronger than NFUB. But perhaps I misunderstand what you have shown. As ever, Bob From solovay@math.berkeley.edu Fri Jun 13 11:04 MDT 1997 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA00633; Fri, 13 Jun 1997 11:04:34 -0600 Return-Path: Received: by math.berkeley.edu (8.7.5/1.33(math)Ow.3) id JAA00812; Fri, 13 Jun 1997 09:56:58 -0700 (PDT) Date: Fri, 13 Jun 1997 09:56:58 -0700 (PDT) From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199706131656.JAA00812@math.berkeley.edu> To: holmes@catseye.idbsu.edu Subject: Re: NFUB's consistency strength Cc: solovay@math.berkeley.edu Status: RO Randall, My connection to the net is somewhat flaky this morning. Let's see if I can get this letter out before it dies again. I tend to agree with Magidor that the characterization [lower bound perhaps] on the consistency of NFU+ should yield [much more than] 0#. The proof would be technical and involve the core model; I don't propose to think through the details myself. So the assumption I am using to get a model of NFUB is much stronger than what you can get from NFUB+. So it goes. I am even more optimistic this morning that I can get a precise equiconsistency result for NFUB [the one conjectured in my last letter]; but I am going to first write up the upper bound before thinking about the lower bound which for various technical reasons looks to be quite tricky. --Bob From holmes@catseye.idbsu.edu Fri Jun 13 13:09 MDT 1997 Received: by catseye.idbsu.edu (1.38.193.4/16.2) id AA00723; Fri, 13 Jun 1997 13:09:44 -0600 Date: Fri, 13 Jun 1997 13:09:44 -0600 From: Randall Holmes Return-Path: To: holmes@catseye.idbsu.edu, solovay@math.berkeley.edu Subject: Re: NFUB's consistency strength Status: RO I have two more communications on this topic. a. an argument for n-subtle (and thus n-ineffable) cardinals in NFUB. b. the other side of the characterization of the strength of NFUB+ in terms of "Morse-Kelley+U". Each of these require a little preparation. They aren't following this note immediately! --Randall From solovay@hotmail.com Fri Jun 13 19:29 MDT 1997 Received: from F50.hotmail.com by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA01109; Fri, 13 Jun 1997 19:29:54 -0600 Return-Path: Received: (from root@localhost) by f50.hotmail.com (8.8.5/8.8.5) id SAA16334; Fri, 13 Jun 1997 18:22:19 -0700 (PDT) Message-Id: <199706140122.SAA16334@f50.hotmail.com> Received: from 207.48.97.205 by www.hotmail.com with HTTP; Fri, 13 Jun 1997 18:22:18 PDT X-Originating-Ip: [207.48.97.205] From: "Robert Solovay" To: holmes@catseye.idbsu.edu Cc: solovay@math.berkeley.edu Subject: The strength of NFUB+ Content-Type: text/plain Date: Fri, 13 Jun 1997 18:22:18 PDT Status: RO Randall, The machine math.berkeley.edu is down, so I am writing from hotmail I just joined this [so as to be able to write you while math.berkeley.edu is down] and I don't plan to use it often. So it's probably best to reply to this address [berkeley may still be down] with a copy to solovay@math.berkeley.edu I'm prepared to announce as a theorem that NFUB+ proves the existence of a Ramsey cardinal kappa with a stationary set of Ramsey cardinals less than kappa. Of course, it follows that NFUB+ proves 0# exists. The proof is really a trivial extension of what you have done. This raises the question "How did Kanamori miss this?" The answer is perhaps that for my trivial proof, I have to go back and do a little more work in NFUB+ [so as to make the measure normal]. Kanamori may have only worked with what you gave him [a model of KM with a predicate for a measure]. I still think that Magidor is right that this [a model of KM with a measure predicate] in itself gives the consequence I stated, but [if one is not allowed to go back and use NFUB+ again] then this probably requires the core-model theory and so is not completely trivial. As ever, Bob --------------------------------------------------------- Get Your *Web-Based* Free Email at http://www.hotmail.com --------------------------------------------------------- From solovay@math.berkeley.edu Fri Jun 13 19:45 MDT 1997 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA01116; Fri, 13 Jun 1997 19:45:33 -0600 Return-Path: Received: by math.berkeley.edu (8.7.5/1.33(math)Ow.3) id SAA18794; Fri, 13 Jun 1997 18:37:58 -0700 (PDT) Date: Fri, 13 Jun 1997 18:37:58 -0700 (PDT) From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199706140137.SAA18794@math.berkeley.edu> To: holmes@catseye.idbsu.edu Subject: Back home. Cc: solovay@math.berkeley.edu Status: RO As you can see, math.berkeley.edu [henceforth "math"] is up again after being down all day. There's no point in sending me the proof re n-ineffables since I have already figured out how to prove much more. [Unless you get it from NFUB rather than NFUB+.] As ever, Bob From solovay@math.berkeley.edu Sun Jun 15 19:15 MDT 1997 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA01466; Sun, 15 Jun 1997 19:15:37 -0600 Return-Path: Received: by math.berkeley.edu (8.7.5/1.33(math)Ow.3) id SAA02883; Sun, 15 Jun 1997 18:08:01 -0700 (PDT) Date: Sun, 15 Jun 1997 18:08:01 -0700 (PDT) From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199706160108.SAA02883@math.berkeley.edu> To: holmes@catseye.idbsu.edu Subject: Retraction Cc: solovay@math.berkeley.edu Status: RO My "proof" that the consistency strength of NFUB lies below 0# has definitely run into trouble and I withdraw that claim as well as the more precise versions that underlay it. There is still a lot of play in the situation, and I intend to continue to think about the question. Since I didn't try to describe my "proof", it's hard to be too precise about the error. Weakly compact cardinals allow one to construct quasi- measures that are not defined on the full power set of kappa but instead on a kappa-sized subset. I used the Erdos cardinal to construct a related series of measures on the various kappa^n's that could be used to construct models with automorphisms. A crucial fact was supposed to be that the only parts of the "iterated ultraproduct' [with index set for the iteration *all the integers* rather than omega] left fixed under the shift automorphism were the constant functions. This is plausible from the analogy with measurable cardinals but seems not to be true in general. In any case, I certainly can't prove it. Sigh! Being infallible is getting harder and harder as the years go by. --Bob From holmes@catseye.idbsu.edu Mon Jun 16 07:35 MDT 1997 Received: by catseye.idbsu.edu (1.38.193.4/16.2) id AA01572; Mon, 16 Jun 1997 07:35:19 -0600 Date: Mon, 16 Jun 1997 07:35:19 -0600 From: Randall Holmes Return-Path: To: holmes@catseye.idbsu.edu, solovay@hotmail.com Subject: Re: The strength of NFUB+ Cc: solovay@math.berkeley.edu Status: RO Certainly KM + measure predicate will give the result itself if my full argument is correct (I claim precise equivalence between this theory and NFUB+). I would like to write up my results about NFUB+; would you be willing to prepare an appendix to go with such a paper proving the result you have stated? I'm hoping to write up the NFUB+ results this summer. --Randall From holmes@catseye.idbsu.edu Mon Jun 16 07:35 MDT 1997 Received: by catseye.idbsu.edu (1.38.193.4/16.2) id AA01583; Mon, 16 Jun 1997 07:35:44 -0600 Date: Mon, 16 Jun 1997 07:35:44 -0600 From: Randall Holmes Return-Path: To: holmes@catseye.idbsu.edu, solovay@math.berkeley.edu Subject: Re: Back home. Status: RO I think that the n-ineffables come from NFUB alone. --Randall From solovay@math.berkeley.edu Mon Jun 16 12:12 MDT 1997 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA01907; Mon, 16 Jun 1997 12:12:40 -0600 Return-Path: Received: by math.berkeley.edu (8.7.5/1.33(math)Ow.3) id LAA24637; Mon, 16 Jun 1997 11:04:59 -0700 (PDT) Date: Mon, 16 Jun 1997 11:04:59 -0700 (PDT) From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199706161804.LAA24637@math.berkeley.edu> To: holmes@catseye.idbsu.edu Subject: Re: The strength of NFUB+ Cc: solovay@math.berkeley.edu Status: RO Randall, I'd be glad to write such an appendix. As I mentioned in a previous letter, I am making a trip to Europe in early July and I plan a trip to Berkeley in August. So those times would be down times. But if I'm right about my argument, it can be pretty brief. --Bob From solovay@math.berkeley.edu Mon Jun 16 12:25 MDT 1997 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA01938; Mon, 16 Jun 1997 12:25:16 -0600 Return-Path: Received: by math.berkeley.edu (8.7.5/1.33(math)Ow.3) id LAA25119; Mon, 16 Jun 1997 11:17:38 -0700 (PDT) Date: Mon, 16 Jun 1997 11:17:38 -0700 (PDT) From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199706161817.LAA25119@math.berkeley.edu> To: holmes@catseye.idbsu.edu Subject: Re: Retraction Cc: solovay@math.berkeley.edu Status: RO Randall, At the moment I'm writing up my repair to my proof of the upper bound on the consistency strength of NFUB. After I do that I want to think about the lower bound. After that, if I'm still here in Eugene, I can write up the Ramsey proof in detail. But basically, one improves the measure result to get a normal measure. Then one is in position to apply the standard arguments to get that the set of Ramsey's less than kappa has measure one. [I hope that choice--the lack thereof for classes--doesn't come up to bite one; I **think** one has enough resources to handle this.] I don't know if anyone has studied KM + a measure before; it certainly feels quite natural to me. It should be the same [consistency strength] as ZF- + V=L[mu] + kappa is the largest cardinal. But there are some subtleties [as I see it] in proving the equivalence having to do with the fact that there is no natural well-ordering of the classes of our starting model of KM. It's hard for me to answer your final question; I don't have a clear grasp on its meaning. Let's put it this way. The definition may be officially new [never explicitly formulated before] but it is *extremely* natural. --Bob From holmes@catseye.idbsu.edu Mon Jun 16 12:58 MDT 1997 Received: by catseye.idbsu.edu (1.38.193.4/16.2) id AA01976; Mon, 16 Jun 1997 12:58:19 -0600 Date: Mon, 16 Jun 1997 12:58:19 -0600 From: Randall Holmes Return-Path: To: holmes@catseye.idbsu.edu, solovay@math.berkeley.edu Subject: Re: Retraction Status: RO Dear Bob, I certainly agree that the definition of KM + measurable predicate is extremely natural! The combination of being extremely natural and officially new makes something interesting in a certain sense, doesn't it? I ran into trouble with the lack of choice on classes; I tried to get a normal measure by building in the proper classes an interpretation of L(kappa+) (in effect) in which one would have choice on proper classes and so be able to construct a normal measure. The lack of choice on proper classes obstructed me, at least, from doing this. Kanamori saw the same problem (though he thought it might be possible to get around it). --Randall From solovay@math.berkeley.edu Mon Jun 16 13:23 MDT 1997 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA02000; Mon, 16 Jun 1997 13:23:26 -0600 Return-Path: Received: by math.berkeley.edu (8.7.5/1.33(math)Ow.3) id MAA27593; Mon, 16 Jun 1997 12:15:48 -0700 (PDT) Date: Mon, 16 Jun 1997 12:15:48 -0700 (PDT) From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199706161915.MAA27593@math.berkeley.edu> To: holmes@catseye.idbsu.edu Subject: Re: Retraction Cc: solovay@math.berkeley.edu Status: RO Randall, In my view, there are things that are so much "in the air" that no particular merit is involved in being the first to write them down. I think that KM + a measure is in this category. YMMV [= "Your mileage may vary"]. --Bob From holmes@catseye.idbsu.edu Mon Jun 16 15:34 MDT 1997 Received: by catseye.idbsu.edu (1.38.193.4/16.2) id AA02060; Mon, 16 Jun 1997 15:34:09 -0600 Date: Mon, 16 Jun 1997 15:34:09 -0600 From: Randall Holmes Return-Path: To: holmes@catseye.idbsu.edu, solovay@math.berkeley.edu Subject: Re: Retraction Status: RO Dear Bob, The only merit of writing it down is in showing that it is equivalent to NFUB+, I suppose. I won't belabor the "originality" of the theory in my treatment, then; I'll say that it is something natural which no one apparently has had any particular reason to formally define. --Randall From solovay@math.berkeley.edu Mon Jun 16 21:12 MDT 1997 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA02218; Mon, 16 Jun 1997 21:12:30 -0600 Return-Path: Received: by math.berkeley.edu (8.7.5/1.33(math)Ow.3) id UAA20590; Mon, 16 Jun 1997 20:04:52 -0700 (PDT) Date: Mon, 16 Jun 1997 20:04:52 -0700 (PDT) From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199706170304.UAA20590@math.berkeley.edu> To: holmes@catseye.idbsu.edu Subject: Retraction: The sequel Cc: solovay@math.berkeley.edu Status: RO Randall, I hereby withdraw my claim that I have shown that NFUB+ entails the consistency of ZFC + "There is a proper class of Ramseys" [or for that matter 0#]. This time the error is easy to state. I wanted to use the assumption that there is no "least non-constant function" to define a decreasing omega sequence of non-constant functions. The problem, as we both have noted is the lack of choice. I planned to use the well-orderings [of some small non-Cantorian rank] in the ambient model. But this doesn't work since the set of codes of functions one would like to pick from is not a set in the ambient NFUB model. Having (a) been burnt by my premature claim re this and (b) needing the result for my projected proof of a lower bound on the consistency of NFUB, I guess I will try to push through the "Magidor" proof and will report if and when I have succeeded. The basic idea is simple. The sets in the coremodel [which are classes of the Cantorian universe] do have a definable well-ordering and one should work with them. There certainly are details to check before one can claim [at least before I should claim] that this works. Possibly relevant is Kunen's old result that the sets constructible from any kappa complete measure on kappa are precisely the sets constructible from a normal measure. Definitely some care is needed since the power set of kappa and the measure itself exist only as virtual objects [sort of like proper classes in ZFC]. --Bob From solovay@math.berkeley.edu Tue Jun 17 00:03 MDT 1997 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA02239; Tue, 17 Jun 1997 00:03:21 -0600 Return-Path: Received: by math.berkeley.edu (8.7.5/1.33(math)Ow.3) id WAA23954; Mon, 16 Jun 1997 22:55:43 -0700 (PDT) Date: Mon, 16 Jun 1997 22:55:43 -0700 (PDT) From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199706170555.WAA23954@math.berkeley.edu> To: holmes@catseye.idbsu.edu Subject: Retracting my retraction Cc: solovay@math.berkeley.edu Status: RO That is, I'm willing to claim once again that NFUB+ proves the consistency of ZFC + "There is a proper class of Ramsey cardinals". My comment that there was a bug in my earlier proof still seems correct to me. The current proof abandons NFUB+ and works solely in the derived theory KM + measure. I had thought that the "Magidor" approach would involve considerations of fine structure and core model theory. But everything is really fairly simple. I will try to outline the proof [up to the point that it joins the correct parts of my earlier proof] in the remainder of this letter. The basic plan is the following. We define an "inner model" of our original model of KM + measure. This inner model satisfies choice so there is no difficulty defining a normal measure from our given measure in "the usual way". And then one can play the "usual games" to get our result. Here is a quick description of the inner model. Form L_\alpha[\nu] where \nu is the measure and \alpha is the least ordinal not of cardinality greater than kappa [the measurable cardinal!] in L[\nu]. It is not quite evident how to make sense of this in KM so I will need to say a bit more. 1. In KM the usual theorems about well-orderings of the ordinals can be proved. Given two such well-orderings, there is a unique isomorphism between one of them and an initial segment of the other. 2. Now let R be a well-ordering of the ordinals. We can form [canonically] a model of V=L[\nu] whose ordinals are order isomorphic to R and which uses the given measure to interpret \nu. Call this model L_R[\nu]. [Its underlying class will be a subset of the constructible sets of our model of KM.] A map of R isomorphically onto an initial segment of S maps L_R[\nu] onto an initial segment of L_S[\nu]. Call a subclass of OR *special* if it appears in some such L_R[\nu]. There is an obvious definable well-ordering of the special classes in order of "their construction in L[\nu]". 3. Suppose that A is special. Let R be awell-ordering of OR of minimal length such that A appears in L_R[\nu]. Then standard arguments show that there is a canonical map [uniformly definable from A] of OR onto the domain of R. [Canonical means that the pull-back of R to a well-ordering of OR depends only on the order-type of R.] Holmes notes: canonical map definable from A of OR onto domain of R? R represents the ordinal alpha such that A first occurs in L_alpha. 4. The collectionof all *special* sets is our model of KM with a definable well- ordering. When we say definable, we mean definable using the restriction of \nu to this model. Standard arguments show that this model is a model of KM [with the predicate for \nu allowed in comprehension axioms.] Roughly, the proof involves showing (a) there are enough special well-orderings to construct all the special sets. (b) We divide into two cases. (1) There is a well-ordering of OR longer than any special well-ordering. Then a failure of comprehension would yield a special well-ordering of length the least ordering which is not special. Holmes adds: failure of comprehension = a special set A which cannot be defined from a special well-ordering. There will be a canonical map (definable from A) of OR onto the domain of R (well-ordering of shortest length yielding A); is that what gives us the special well-ordering of a set with no special well-ordering? The special set itself can be well-ordered suitably, giving a special well-ordering? (2) Otherwise. Then a failure of comprehension would yield a new well-ordering of the ordinals longer than any special ordering contrary to our current case. Holmes adds: failure of comprehension = a special set A which cannot be defined from a special well-ordering. If there are no non-special well-orderings, this seems obviously ruled out? Many details are of course omitted in the above. The way to fill them in is by imitating familiar arguments from tje easy parts of Jensen's fine structure paper. [E.g. if a new subset of omega appears in L at stage alpha then alpha is countable in L.] So now we have an inner model of KM + measure where there is a well-ordering definable from \nu. Now it is easy to do the "least function" trick and get another such model where \nu is normal. I suspect [but **have not checked**] that the Kunen result PLEASE IGNORE THE IMMEDIATELY PRECEDING LINE! As ever, Bob From holmes@catseye.idbsu.edu Wed Jun 18 15:39 MDT 1997 Received: by catseye.idbsu.edu (1.38.193.4/16.2) id AA03548; Wed, 18 Jun 1997 15:39:50 -0600 Date: Wed, 18 Jun 1997 15:39:50 -0600 From: Randall Holmes Return-Path: To: holmes@catseye.idbsu.edu, solovay@math.berkeley.edu Subject: Re: Silence? Status: RO Dear Bob, I've been busy and have not yet examined the argument in detail. I do understand the general idea of what you are doing. I tried to do exactly this, but I couldn't convince myself that the resulting structure was a model of KM. I will report within a day or two whether I am now convinced; I probably would benefit from an expanded argument. I find this: "\alpha is the least ordinal not of cardinality greater than kappa [the measurable cardinal!] in L[\nu]." a little confusing; wouldn't 0 fulfil this definition? I was hoping to figure out from context what you meant; I haven't yet had time to sit down and read this letter in detail. I have not read Jensen's fine structure paper; this would doubtless be very good for me! I do understand the least function trick for getting a model with a normal ultrafilter; I was trying to build this same interpretation of L in the proper classes to get exactly that. --Randall From solovay@isiax1.isi.it Wed Jul 9 01:08 MDT 1997 Received: from [130.192.70.11] by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA13688; Wed, 9 Jul 1997 01:08:19 -0600 Return-Path: Received: by isiax1.isi.it; id AA05952; Wed, 9 Jul 1997 10:02:34 +0200 Date: Wed, 9 Jul 1997 10:02:33 +0200 (MET DST) From: Solovay Robert To: Randall Holmes Subject: Preliminary announcement Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO Randall, let me get all the disclaimers taken care of first. The result I am going to mention is very new [less than two hours old] though it's been in gestation for more than a month. One of the directions is quite trick and complicated and I'll only be sure that it's right after I've written up a substantial portion of the proof. [A process which I am going to begin almost immediately.] Also please note that until July 18th the best email address for me is solovay@isiax1.isi.it NB The character directly preceding the first dot is a "one" and not an "ell". After July 18th, the best email address for me will again revert to solovay@math.berkeley.edu So here's the result: The following two theories are equiconsistent: 1) NFUB 2) ZFC- + "There is a weakly compact cardinal". The direction from 1) to 2) is rather easy [though it does involve some Jensenlehre to go from a model of KM to one of KM + V=L]. The direction from 2) to 1) is quite tricky and in particular the use of the weakly compact cardinal is rather subtle. Of course, we would be in a rather embarassing situation if you had not withdrawn your claim that NFUB proves the existence of n-subtles. I'm rather pleased with this outcome since I feared that an elegant equiconsistency result was not in the cards. As ever, Bob From holmes@catseye.idbsu.edu Wed Jul 9 08:46 MDT 1997 Received: by catseye.idbsu.edu (1.38.193.4/16.2) id AA13788; Wed, 9 Jul 1997 08:46:19 -0600 Date: Wed, 9 Jul 1997 08:46:19 -0600 From: Randall Holmes Return-Path: To: holmes@catseye.idbsu.edu, solovay@isiax1.isi.it Subject: Re: Preliminary announcement Status: RO Dear Bob, I do not have a result about n-subtles at this point; this result about NFUB is one which I have always regarded as possible, but whose proof I was quite certain would have to be very tricky. Where in Italy are you? I hope that you will have a chance to look at my argument about n-Mahlos in NFUA; I'm pretty sure that it works, and it extends the "descent" idea of your inaccessibles proof neatly. The essential point of that proof is that your argument can be adapted to show that any club which is downward natural (in which min(x,T(x)) is an element if max(x,T(x)) is an element) contains an inaccessible (this bootstraps to "each such club contains an n-Mahlo" in the course of the proof); the way this is done is to compose the operations in your descent argument for inaccessibles with the operation "descend to the largest element of the club at or below this point" (this commutes with T by downward naturality of the club). This result can be used to show that the first ordinal index at which suitably chosen clubs without inaccessibles (resp. n-Mahlos) below inaccessibles (resp. n-Mahlos) alpha and T(alpha) differ is cantorian, which eliminates the bad case in the descent argument. The condition "every club which is downward natural contains an n-Mahlo" is not suitable for (internal) induction on n, being unstratified, so an unfortunate generalization of the result is not going to occur! I think that my remark in the original note about the need to use n+1 iterated images under the T operation is not a correct explanation of why an internal induction does not work. The argument can be set up with pairs of cardinals at every stage, I think. --Randall P.S. I assume that you mean "ZFC - Power Set" by ZFC- P.P.S. Forster and I will be reviewing the inaccessibles argument as presented in the book at the end of July, when I will be at Cambridge for the NF 60th anniversary meeting; after that the book will be in final form. I did put the comment in the notes that your own proof was carried out in (a fragment of) ZFC with an external automorphism. Thank you for pointing out the problem with my original presentation (not to mention the stupid misstatement of the collapsing lemma; I really do know better!) P.P.P.S. The following property of cardinals appears to allow construction of models of NFUB+. Is it a familiar one? Let kappa be a cardinal such that there is a nonempty class Q of unbounded sets W of kappa such that for any class W \in Q, and any function S from the set of all finite subsets of kappa to kappa which takes a finite subset F to a subset of min F in all cases, there are sets S_n for each n and a V \subseteq W such that V \in Q and S(A) = S_{card F} \cap min F for any finite subset F of V. I obtained this by taking a version of ineffability applying to finite sets of all sizes and extending it to a strong "complete ineffability". It is clearly at least as strong as a Ramsey; presumably stronger. From solovay@isiax1.isi.it Wed Jul 9 11:05 MDT 1997 Received: from [130.192.70.11] by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA13864; Wed, 9 Jul 1997 11:05:17 -0600 Return-Path: Received: by isiax1.isi.it; id AA06568; Wed, 9 Jul 1997 19:58:49 +0200 Date: Wed, 9 Jul 1997 19:58:48 +0200 (MET DST) From: Solovay Robert To: Randall Holmes Subject: Reply to your letter of July 9th Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO Randall, I am in Turin, Italy attending a conference on "Quantum Computation". Whether I will be able to look at your proo on n-Mahlos is quite problematic. I am desparate to get some papers out, and I plan to start the process as soon as I have checked [by writing up a preliminary draft] the results on NFUB. Relative to my priorities, I have spent more time on NFish issues than I had planned to and I also want to get my "learning physics" project [as well as some other projects] back into gear. I do plan to write up my results on NFUA and NFUB for publiction, but I lack the time to debug the proofs of other people. Yes, by ZFC- I mean ZFC minus the power set axiom. An equivalent formulation is KM [including the global axiom of choice for sets] together with an assertion that "The class of ordinals is weakly compact". [That is, the two theories have the same consistency strength.] I chose the ZFC- version for stating the result since the meaning of the sentence in quotes concerning OR is perhaps unclear. If your property allows the construction of models of NFUB+ it **must** be much stronger than being Ramsey. Offhand, it didn't ring any bells. I suspct [but have not verified] that NFUB+ is equiconsistent with your formulation of KM "with a measure on OR". Of course, the direction from NFUB to the strong form of KM is a theorem of yours; I forget whether you've claimed the reverse direction. As ever, Bob From solovay@isiax1.isi.it Wed Jul 9 11:05 MDT 1997 Received: from [130.192.70.11] by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA13864; Wed, 9 Jul 1997 11:05:17 -0600 Return-Path: Received: by isiax1.isi.it; id AA06568; Wed, 9 Jul 1997 19:58:49 +0200 Date: Wed, 9 Jul 1997 19:58:48 +0200 (MET DST) From: Solovay Robert To: Randall Holmes Subject: Reply to your letter of July 9th Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO Randall, I am in Turin, Italy attending a conference on "Quantum Computation". Whether I will be able to look at your proo on n-Mahlos is quite problematic. I am desparate to get some papers out, and I plan to start the process as soon as I have checked [by writing up a preliminary draft] the results on NFUB. Relative to my priorities, I have spent more time on NFish issues than I had planned to and I also want to get my "learning physics" project [as well as some other projects] back into gear. I do plan to write up my results on NFUA and NFUB for publiction, but I lack the time to debug the proofs of other people. Yes, by ZFC- I mean ZFC minus the power set axiom. An equivalent formulation is KM [including the global axiom of choice for sets] together with an assertion that "The class of ordinals is weakly compact". [That is, the two theories have the same consistency strength.] I chose the ZFC- version for stating the result since the meaning of the sentence in quotes concerning OR is perhaps unclear. If your property allows the construction of models of NFUB+ it **must** be much stronger than being Ramsey. Offhand, it didn't ring any bells. I suspct [but have not verified] that NFUB+ is equiconsistent with your formulation of KM "with a measure on OR". Of course, the direction from NFUB to the strong form of KM is a theorem of yours; I forget whether you've claimed the reverse direction. As ever, Bob From solovay@isiax1.isi.it Wed Jul 9 13:25 MDT 1997 Received: from [130.192.70.11] by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA14033; Wed, 9 Jul 1997 13:25:17 -0600 Return-Path: Received: by isiax1.isi.it; id AA06749; Wed, 9 Jul 1997 22:19:35 +0200 Date: Wed, 9 Jul 1997 22:19:35 +0200 (MET DST) From: Solovay Robert To: Randall Holmes Subject: Re: Reply to your letter of July 9th In-Reply-To: <9707091924.AA06750@isiax1.isi.it> Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO On Wed, 9 Jul 1997, Randall Holmes wrote: > Dear Bob, > > I do claim the other direction (on NFUB+ = KM with a measure on OR); > one builds the kind of ill-founded iterated ultrapower that one constructs > to get a model of NFUB+ from a measurable; the work (straightforward but > tedious) is in showing that everything can be coded in KM with a measure > on OR. > > I will write up the n-Mahlo proof in detail and send it to you, but I > will regard it as purely serendipitous if I get a response. How about > that? Part of my agenda for this summer is to write up the > equiconsistency of NFUB+ and KM + measure on the ordinals; if I don't > come up with my own proof for Ramsey cardinals in this theory, I'll > ask you for an appendix on the subject (as we already discussed). > This may get shoved into early fall; my top priority right now is to > write papers on my theorem proving work. > > --Randall > It's always fine to send me things. If you need an appendix I will provide one. --Bob From solovay@isiax1.isi.it Fri Jul 18 11:54 MDT 1997 Received: from isiax1.isi.it by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA20772; Fri, 18 Jul 1997 11:54:42 -0600 Return-Path: Received: by isiax1.isi.it; id AA14703; Fri, 18 Jul 1997 20:48:41 +0200 Date: Fri, 18 Jul 1997 20:48:41 +0200 (MET DST) From: Solovay Robert To: Randall Holmes Subject: The proof has survived Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO Randall, A bunch of things. 1. I've now thoroughly checked my equiconsistency result. There is a written document that corresponds to the check that I will be mailing to you tonight. 2. Tomorrow morning I begin the two day trek back to Oregon. So you should revert to my usual email address: solovay@math.berkeley.edu 3. I'm going to post the draft of the paper to Mitchell's email list approx. July 31st [assuming the proof doesn't die in the meantime]. I think I've been fair to your prior work, but feedback is welcomed. Also feedback on the correctness and or clarity of the proof is welcomed, though clarity issues probably won't be explicitly addressed till I write the second draft. [There are at least three papers ahead of draft 2. Draft 2 should also include my results on NFUA.] At the same time that I post to Mitchell's list, I'll send an announcement to the NF mailing list. 4. Since many mail programs mangle Latex files, I have gzipped and uuencoded the parts of the paper. Later tonight [after dinner] I will send you three letters with subjects equi2.uu, part1.uu and part2.uu I believe the following instructions should convert them to a printable paper: Make a temporary directory [say build_it ] and store the forthcoming three letters there. %cd build_it %uudecode *.uu #Assuming you name them according to their subject lines %gunzip *.gz %latex equi2.tex %latex equi2.tex Now print up equi2.dvi the way it's done on your local system. If this rigarmarole fails to work, I'll just email the raw latex files. The proof turned out to simplify while I was writing it up. It's still not completely trivial, but it breaks nicely into "bite-sized pieces". As ever, Bob P. S. As per usual, if you decide to read the proof, and any part is obscure, I'll be glad to [try to] clarify it. I just won't get into the reqriting of this draft until other papers are done. From solovay@math.berkeley.edu Thu Feb 26 15:11 MST 1998 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA20399; Thu, 26 Feb 1998 15:11:40 -0700 Return-Path: Received: (from solovay@localhost) by math.berkeley.edu (8.8.7/8.8.7) id OAA26980; Thu, 26 Feb 1998 14:07:25 -0800 (PST) Date: Thu, 26 Feb 1998 14:07:25 -0800 (PST) From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199802262207.OAA26980@math.berkeley.edu> To: holmes@catseye.idbsu.edu Subject: Your recent query Cc: solovay@math.berkeley.edu Status: RO %% %% Randall, %% I presume that you will have no trouble printing up this letter. If %% for any reason, you do have trouble, let me know and I'll mail you %% hard copy. [Also, you probably can just read the source code to get %% most of the meat of the letter %% \documentclass[12pt]{article} \usepackage{amstex} \begin{document} \thispagestyle{empty} \begin{flushright} PO Box 5949\\ Eugene, OR 97405\\ February 26, 1998 \end{flushright} \begin{flushleft} Dear Randall, \end{flushleft} \section{} Let me start with the reference to Jensen's paper: Ronald B. Jensen, {\em The fine structure of the constructible hierarchy.} Annals of Mathematical Logic \textbf{4} (1972), 229-308. The journal ``Annals of Mathematical Logic'' did a name change to ``Annals of Pure and Applied Logic'' in 1983 and the journal may well be filed in your library under the latter title [including volume 4 of AML]. Jensen uses a slightly different hierarchy for $L$, the $J_\alpha$'s. The results I assert for the $L_\alpha[\nu]$'s are correct as they stand [though I don't know a really satisfactory reference]. Alternatively, one could rework them in terms of a suitable version of $J_\alpha[\nu]$. I remark also that I am not actually thinking in what follows about the measure $\nu$ [a function with domain the powerset of $OR$ and taking values in $\{0,1\}$] but rather the associated collection $D$ of sets of measure $1$. I doubt that this makes a difference, but it is a shade easier to think in terms of $D$. Jensen proves much more than is needed for the arguments I will sketch. But I don't know any reference which does ``just enough''. \section{} Many of the remarks you make are right on the money. One has to learn to do ``model theory'' inside the $L_\alpha[D]$'s. This can be done, but I don't know any elegant description of the set theory that obtains in these models. By eithery working in $L_\alpha[D]$'s for $\alpha$ of limit order type or by working in the $J_\alpha[D]$'s one can assume that they are closed under the usual ordered pair. And the fact that $L_{\alpha + 1}[D]$ is the collection of sets first order definable in the structure \[\langle L_\alpha[D] \mid \in, D \cap L_\alpha[D]\] can play the role of comprehension. \section{} I come now to my outline as presented in your second letter to me. \subsection{} First a minor correction. I write A map of $R$ onto an initial segment of $S$ maps $L_R[\nu]$ onto an initial segment of $L_S[\nu]$. I would now phrase this as follows: A map of $R$ onto an initial segment of $S$ determines a map of $L_R[D]$ onto a ``transitive'' subcollection of $L_S[D]$. The quotes around ``transitive'' are because the ``epsilon relation'' of $L_S[D]$ is not literally the usual $\epsilon$ relation. \subsection{} I now provide further details on the construction of the canonical map discussed in point 3. Recall the situation. $A$ is a special subset of $OR$. This means that for some well-ordering $R$ of $OR$, $A$ appears in $L_R[D]$. We take $R$ to be as short as possible so that $A$ appears in $L_R[D]$. We introduce some slightly longer well-orderings of $OR$: $R_1$ will have the order type of ``$R$ followed by $\omega$''. $R_2$ will have the order type of ``$R_1$ followed by $OR$''. $R_3$ will have the order type of ``$R_2$ followed by $\omega$''. We will eventually show the following: \begin{enumerate} \item In $L_{R_1}[D]$, there is a map $\Psi$ of $OR$ onto $L_R[D]$. \item In $L_{R_3}[D]$ there is a well ordering of $OR$, $R^\star$, of the same order-type as $R$ \end{enumerate} \subsection{} It will ease the notation somewhat to work ``inside'' the model $L_{R_1}[D]$. In this model there are the following important ordinals and sets: \begin{enumerate} \item $\kappa$. This is the order type of $OR$. \item $\eta$. This is the order type of $R$. \item $L_\eta[D]$. \item $<_1$. This is the usual ``canonical well-ordering'' of $L_\eta[D]$. \item $A$. [Our ``special'' set.] \item The following structure ${\cal M}$: The underlying set of ${\cal M}$ will be $L_\eta[D]$. The structure will have the following predicates: \begin{enumerate} \item The restriction of the $\epsilon$ relation to $L_\eta[D]$. \item $D$ [viewed as a one-place predicate]. \item $<_1$. \item $A$ [viewed as a constant] \end{enumerate} \end{enumerate} We now define an elementary submodel of ${\cal M}$, namely the Skolem hull of $\kappa$ in ${\cal M}$. Let us call the resulting structure ${\cal M}'$. \subsection{} I temporarily emerge from $L_{R_1}[D]$ to our original model of $KM$. The model ${\cal M}'$ is isomorphic to a model of the form $L_S[D]$ where $S$ is the restriction of $R$ to those elements of the domain of $R$ that ``appear in as ordinals in ${\cal M}'$''. It is evident that $A$ appears in $S$ and that $S$ has order type $\leq$ to that of $R$. By the minimality of $R$ we conclude that $S$ has the same order-type of $R$. Thus ${\cal M}'$ is isomorphic to ${\cal M}$. But ${\cal M}'$ is also an elementary substructure of ${\cal M}$ and it has the property that every element is definable from some ordinal less than $\kappa$. So this property is shared by the isomorphic copy ${\cal M}$. We now see that ${\cal M}'$ is not just isomorphic to ${\cal M}$ but {\em identical} to ${\cal M}$. \subsection{} We now return to the model $L_{R_1}[D]$. We define a map $\Psi_1$ from $\omega \times \kappa$ {\em onto} $L_\eta[D]$ as follows: Let $\langle i, \xi\rangle$ be an element of $\omega \times \kappa$. Suppose first that [the interesting case] \begin{enumerate} \item $i$ is the G\"{o}del number of a formula $\psi(x,y)$ of the language appropriate to the structure ${\cal M}$ having the indicated free variables. \item The following holds in ${\cal M}$: For exactly one $x$, $\psi(x,\xi)$. \end{enumerate} Then $\Psi_1(i,\xi)$ is the unique $x \in L_\eta[D]$ such that $\psi(x,\xi)$ holds in ${\cal M}$. If we are not in the interesting case, $\Psi_1(i,\xi) = 0$. \subsection{} It is easy to massage $\Psi_1$ to a map $\Psi$ which maps $\kappa$ onto $L_\eta[D]$. $\Psi$ will also lie in $L_{R_1}[D]$. Define a subset $E$ of $\kappa$: $\gamma \in E$ iff for some $\lambda < \eta$, $\gamma$ is the least ordinal such that $\Psi(\gamma) = \lambda$. It is evident that $E$ has order type $\kappa$. The unique order isomorphism $h : \kappa \mapsto E$ will lie in $L_{R_3}[D]$. Working now in $L_{R_3}[D]$ we define a well-ordering $R^\star$ of $\kappa$ of order type $\eta$: $\xi_1 R^\star \xi_2$ iff $\Psi(h(\xi_1)) < \Psi(h(\xi_2))$. \subsection{} I have now completed the task I set out to do. Let me know if you have any questions [or indeed, if you {\em don't} have any questions]. \end{document} From solovay@math.berkeley.edu Fri Feb 27 13:44 MST 1998 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA21720; Fri, 27 Feb 1998 13:44:27 -0700 Return-Path: Received: (from solovay@localhost) by math.berkeley.edu (8.8.7/8.8.7) id MAA02310; Fri, 27 Feb 1998 12:40:08 -0800 (PST) Date: Fri, 27 Feb 1998 12:40:08 -0800 (PST) From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199802272040.MAA02310@math.berkeley.edu> To: holmes@catseye.idbsu.edu Subject: Getting models of KM + V=L Cc: solovay@math.berkeley.edu Status: RO Randall, Your annotations on my argument [getting a model of KM +V=L from a model of KM] make me think that perhaps you do not understand a crucial step. First I quote from your annotated version of my argument and then I give some further comments on the argument I have in view. [My discussion in the quoted material concerns L[nu] where nu is a measure on OR. But the issues are entirely analogous for L and in the new material I will consider that case.] \begin{quote} Roughly, the proof involves showing (a) there are enough special well-orderings to construct all the special sets. (b) We divide into two cases. (1) There is a well-ordering of OR longer than any special well-ordering. Then a failure of comprehension would yield a special well-ordering of length the least ordering which is not special. Holmes adds: failure of comprehension = a special set A which cannot be defined from a special well-ordering. There will be a canonical map (definable from A) of OR onto the domain of R (well-ordering of shortest length yielding A); is that what gives us the special well-ordering of a set with no special well-ordering? The special set itself can be well-ordered suitably, giving a special well-ordering? (2) Otherwise. Then a failure of comprehension would yield a new well-ordering of the ordinals longer than any special ordering contrary to our current case. Holmes adds: failure of comprehension = a special set A which cannot be defined from a special well-ordering. If there are no non-special well-orderings, this seems obviously ruled out? This remark by Holmes is obviously wrong-headed; what we are looking at is a non-special collection definable by quantifications involving all special sets. \end{quote} The remarks you make on case (a) seem ok. If R is the least well-ordering of OR longer than any special well-ordering of OR, then a failure of comprehension would yield that R is in fact special [as in the letter I sent you yesterday]. But the treatment of case (b) [all well-orderings of R are special] requires a tad more. Just as in the argument yesterday, one constructs [from the failure of comprehension] a map of OR onto the totality of all special orderings. Using this map, one constructs a *new* well-ordering of OR which is longer than any given well-ordering of OR [a contradiction!] Note that the failure of comprehension does not immediately yield a new special set since special sets are those lying in L_R for R some well-ordering of OR. The set A defined by failure of comprehension does not [by assumption] lie in such an L_R. I hope this is more illuminating than confusing. --Bob A final remark. Here instead of a full elementary submodel of a class-sized structure, one is constructing a Sigma_n elementary structure of a larger than class-sized structure ["the union of all the L_R's"]. Notes to myself: The fact that we are working on the superclass ordinal is disconcerting! In this case, we are considering a set A which would be introduced at L_[kappa+ + 1] (the superclass ordinal plus one). Our strategy is to build a hull for L[kappa+]. The definition of A gives us a new subset of kappa which would be added at L[kappa+]. Now build a hull for L[kappa+] using the elements of kappa, the set A, and the facts about L[kappa+]. What logical limitations are encountered? We need to build only enough of a hull to get the definition of A to work? From solovay@math.berkeley.edu Fri Feb 27 17:25 MST 1998 Received: from math.Berkeley.EDU by catseye.idbsu.edu with SMTP (1.38.193.4/16.2) id AA21971; Fri, 27 Feb 1998 17:25:58 -0700 Return-Path: Received: (from solovay@localhost) by math.berkeley.edu (8.8.7/8.8.7) id QAA10538; Fri, 27 Feb 1998 16:21:42 -0800 (PST) Date: Fri, 27 Feb 1998 16:21:42 -0800 (PST) From: solovay@math.berkeley.edu (Robert M. Solovay) Message-Id: <199802280021.QAA10538@math.berkeley.edu> To: holmes@catseye.idbsu.edu Subject: Your letter on L. Cc: solovay@math.berkeley.edu Status: R I think of things very slightly differently but perhaps the difference doesn't matter. Literally, I wasn't forming the hull, just working with the "coded version" [which is a well-defined class]. The reason one has to restrict to Sigma_n hulls for some large n is so that the definition of this coded version is a class. One now has that this "new class A" is special after all. I certainly agree that I didn't spell this all out in the prior letter. So the short answer is that your approach is viable provided you don't literally try to define the hull. [Of course, just as in ZFC one can sort of speak of classes, in KM one can sort of speak of hyperclasses.] I guess the upshot is that I am having a hard time decidng if what you say is "right on the nose". But the mathematics is clear enough. --Bob From solovay@math.berkeley.edu Sat Apr 22 00:49:25 2000 Return-Path: Received: from diamond.idbsu.edu (root@diamond.idbsu.edu [132.178.200.127]) by catseye.idbsu.edu (8.8.7/8.8.7) with ESMTP id AAA04669 for ; Sat, 22 Apr 2000 00:49:25 -0600 Received: from math.berkeley.edu (gold-slow.Math.Berkeley.EDU [128.32.183.94]) by diamond.idbsu.edu (8.9.3/8.9.3) with ESMTP id AAA09253 for ; Sat, 22 Apr 2000 00:46:49 -0600 Received: from blue2.math.Berkeley.EDU (blue2.math.berkeley.edu [169.229.58.60]) by math.berkeley.edu (8.9.3/8.9.3) with ESMTP id XAA15993; Fri, 21 Apr 2000 23:46:45 -0700 (PDT) From: "Robert M. Solovay" Received: (from solovay@localhost) by blue2.math.Berkeley.EDU (8.9.3/8.9.3) id XAA12043; Fri, 21 Apr 2000 23:46:54 -0700 (PDT) Date: Fri, 21 Apr 2000 23:46:54 -0700 (PDT) Message-Id: <200004220646.XAA12043@blue2.math.Berkeley.EDU> To: holmes@diamond.idbsu.edu Subject: NFU*: Letter1 Cc: solovay@math.berkeley.edu Status: RO Randall, Here starts my series of letters about my results on NFU*. I will follow the same conventions as I did in my presentation about the proofs concerning NFUA. That is, it will take me a series of letters. How much I gewt to in a particular letter will depend on when my time and energy runs out. There is also the usual problem of saying too little [or too much]. If I say something is obvious or well-known and and it's not for you, **please** let me know. At some point, I may start transmitting these letters as some incarnation of TeX file (e.g. dvi files or ps files). I'm in the process of changing the detailed way I produce TeX files and I need to experiment with various macro packages. If I try them on this series of letters, I can kill two birds with one stone. Of course, if it turns out to be too much hassle for you to print up the TeX stuff, I can revert to ascii letters. Let's start by being sure we are talking about the same things: 1) By NFU* I mean the theory NFU [including choice and infinity] together with the following additional axioms: (a) The axiom of counting: omega is strongly cantorian. (b) Selection for strongly cantorian sets: If x is strongly cantorian, and phi is a formula [no stratification restrictions imposed and parameters allowed], then the set of all y in x such that phi(y) exists. [Of course, this is a scheme of axioms.] 2) Next I want to describe the ZFish theory which will be proved to have the same consistency strength: (a) First we will have ZC. This is all the axioms of ZFC except the replacement axiom. In particular, it has the selection scheme for arbitrary formulas. (b) Then we will have replacement for formulas that are Sigma_2 in the Levy hierarchy. Since we have full selection, we don't have to worry about the issue of the domain of the function under consideration. We get an equivalent version of the axiom if we require that the domain of the function in question is an ordinal. [To get things started if we do this, we should throw in an axiom giving Mostowski collapse, so that we have the usual theory of Von Neumann ordinals at hand.] It is true that Sigma_2 replacement can be expressed as a finite set of axioms. This isn't important for us, however. What is important is that the theory described in 2) holds in L if it holds in V. Also if it holds in V and B is a complete Boolean algebra, then it holds in V^B. I view these results as "well-known". They are not completely trivial, however. So the main theorem is that NFU* is consistent iff ZC + Sigma_2 replacement is consistent. The proof can be formalized in PRA [primitive recursive arithmetic]. But I won't insist on this, and all I am officially claiming is that the equiconsistency proof can be carried out in second order aritmetic [or what comes to the same thing, ZF-]. As seems to be the custom in this sort of thing, the two directions of the equiconsistency are by totally different arguments. The direction getting Con(ZC + Sigma_2 replacement) from Con(NFU*) is quite a bit easier, and I will start [and probably finish] that proof in the next letter. One final remark. I expected the consistency strength of NFU* to be much stronger than it turned out to be. On first glance, it looks rather similar to NFUB. This ends letter #1. To be continued ... --Bob From solovay@math.berkeley.edu Sat Apr 22 20:28:10 2000 Return-Path: Received: from diamond.idbsu.edu (root@diamond.idbsu.edu [132.178.200.127]) by catseye.idbsu.edu (8.8.7/8.8.7) with ESMTP id UAA05675 for ; Sat, 22 Apr 2000 20:28:10 -0600 Received: from math.berkeley.edu (gold.Math.Berkeley.EDU [169.229.58.61]) by diamond.idbsu.edu (8.9.3/8.9.3) with ESMTP id UAA29537 for ; Sat, 22 Apr 2000 20:25:30 -0600 Received: from blue2.math.Berkeley.EDU (blue2.math.berkeley.edu [169.229.58.60]) by math.berkeley.edu (8.9.3/8.9.3) with ESMTP id TAA07640; Sat, 22 Apr 2000 19:25:29 -0700 (PDT) From: "Robert M. Solovay" Received: (from solovay@localhost) by blue2.math.Berkeley.EDU (8.9.3/8.9.3) id TAA16681; Sat, 22 Apr 2000 19:25:39 -0700 (PDT) Date: Sat, 22 Apr 2000 19:25:39 -0700 (PDT) Message-Id: <200004230225.TAA16681@blue2.math.Berkeley.EDU> To: holmes@diamond.idbsu.edu Subject: NFU*: Letter 2 Cc: solovay@math.berkeley.edu Status: RO We are given a model of NFU*. Our goal is to construct a model of ZC + Sigma_2 replacement. Let the given model of NFU* be M, with epsilon relation epsilon_M. Let Z be as usual the set of isomorphism classes of extensionial topped well-founded relations. We write epsilon_Z for the epsilon relation on Z. We define a subclass of Z, W as follows: An equivalence class z in Z will lie in W, if the underlying set of any representative is strongly cantorian. W is, in the obvious sense, a transitive subclass of Z. We equip W with the epsilon relation which is the restriction of epsilon_Z to W. We shall show that W is a model of ZC + Sigma_2 Replacement. Because of the many epsilon relations floating around, I am taking a little license in describing things. Thus, really, epsilon_W is an ordinary binary relation on W. And W is really a subset [in the sense of the metatheory] of the [ordinary] set whose members are those things that M thinks are members of Z. But I shall ignore such fine points for the most part. It is quite straightforward to see that W is a model of ZC. The axiom of infinity holds in W since the axiom of counting holds in M. And the replacement schema holds in W since "replacement for strongly Cantorian sets" holds in M. You mean "selection" not "replacement" here, don't you? We know that for any ordinal alpha of W, Beth_alpha exists in Z. Using induction on alpha, we can see that in fact for any alpha in W, Beth_alpha exists in W. Recall that an ordinal alpha is a Beth fixed point [we will abreviate this bfp] if alpha = Beth_alpha. Using full induction on omega [which is available in NFU*] and the result of the preceding paragraph, it is easy to see that for any alpha in W, there is a bfp beta in W which is greater than alpha. An ordinal alpha is a limit of Beth fixed points [we abreviate this lbfp] if it is greater than 0 and for any eta < alpha, there is a bfp beta with eta < beta < alpha. Clearly any lbfp is a bfp. Again, it is easy to see that if alpha is in W, there is a lbfp beta in W with alpha < beta. Note that it is now clear [since W is an initial segment of Z which is a model of ZFC-] that the Mostowski collapse theorem holds in W. [In fact, it was clear at the instant that W was defined.] We next need to recall the Levy collapse lemma [which we view as a theorem of ZFC-: Let kappa be an uncountable cardinal. Let H(kappa) be the collection of all sets whose transitive closure has cardinality less than kappa. Then if the parameters of a Sigma_1 formula phi(x) lie in H(kappa) and there is an x such that phi(x). then there is an x in H(kappa) such that phi(x) [and conversely]. We express this by saying that H(kappa) is absolute for Sigma_1 formulas. Now an uncountable cardinal kappa is a bfp iff V_kappa = H(kappa). It follows that if beta is a bfp, then beta is absolute for Sigma_1 formulas. We now return to the task of proving that Sigma_2 replacement holds in W. Let G be a Sigma_2 function with domain an ordinal delta in W. We have to show that the range of G is a member of W. We introduce an auxilliary function H. H(alpha) is defined iff alpha < delta. If so, H(alpha) is the least beta such that: (1) beta is a bfp (2) The Sigma_2 definition of G(alpha) works in V(beta) to define G(alpha). (3) beta is the least ordinal satisfying (1) and (2). The following points should be clear: (1) In W, H is defined on all of delta. (2) The same definition works in Z to define a function on delta. And it defines the same function in Z that it does in W. But Z is a model of ZFC-. So the sup of the range of H exists in Z. But it is the sup of strongly cantorian ordinals, hence itself strongly cantorian. That is, the range of H is bounded in W. It is now clear that the range of G is a set in W. This completes our verification that Sigma_2 Replacement holds in W and with that our proof that Con(NFU*) entails Con(ZC + Sigma_2 Replacement). This ends letter 2. --Bob From solovay@math.berkeley.edu Sun Apr 23 21:42:47 2000 Return-Path: Received: from diamond.idbsu.edu (root@diamond.idbsu.edu [132.178.200.127]) by catseye.idbsu.edu (8.8.7/8.8.7) with ESMTP id VAA06980 for ; Sun, 23 Apr 2000 21:42:46 -0600 Received: from math.berkeley.edu (gold-slow.Math.Berkeley.EDU [128.32.183.94]) by diamond.idbsu.edu (8.9.3/8.9.3) with ESMTP id VAA04871 for ; Sun, 23 Apr 2000 21:40:06 -0600 Received: from blue2.math.Berkeley.EDU (blue2.math.berkeley.edu [169.229.58.60]) by math.berkeley.edu (8.9.3/8.9.3) with ESMTP id UAA08827; Sun, 23 Apr 2000 20:40:03 -0700 (PDT) From: "Robert M. Solovay" Received: (from solovay@localhost) by blue2.math.Berkeley.EDU (8.9.3/8.9.3) id UAA23634; Sun, 23 Apr 2000 20:40:15 -0700 (PDT) Date: Sun, 23 Apr 2000 20:40:15 -0700 (PDT) Message-Id: <200004240340.UAA23634@blue2.math.Berkeley.EDU> To: holmes@diamond.idbsu.edu Subject: NFU*: The hard direction--preliminary outline Cc: solovay@math.berkeley.edu Status: RO We turn now to the reverse direction of our equiconsistency result. We are going to present three results which will converge to the final proof. A) Work in ZFC + V=L + "There is an inaccessible cardinal". There is a model of NFU*. [In fact this proof is easily modified to get a model whose strongly cantorian sets are a model of ZFC.] Question: Is the strength of NFU + Choice + Counting + Replacement for s.c. sets the same as the strength of ZFC? The proof of A) is modeled on Jensen's proof that for any ordinal alpha, there is a model of NFU whose standard part is alpha. Like Jensen's proof the model is constructed in a length omega construction. Unlike Jensen's proof, the value of alpha is not known in advance. Instead it is dynamically generated by the construction. During the course of the construction we will consider terms that denote Cantorian ordinals. We will take steps to insure that one of the following alternatives happens: (a) The value of the term is less than alpha; (b) There is a non-Cantorian ordinal whose value is less than the value of the term. We could easily arrange that the proof of part A takes place in ZFC [rather than ZFC + "There is an inaccessible" + V=L]. Part B is obtained by optimizing part A: We prove in ZF- that if there is a model M of ZC + Sigma_3 Replacement then there is a model of V=L. There are various technical complications involved in the transition from part A to part B: 1) The model M need not be an omega model. This causes no real problems, but it requires all constructions to be done "internal to M". 2) What plays the role of alpha is now the class of ordinals of M. The model of NFU* we construct is a proper class of M. In part C, we prove our final result [in ZF-]: If there is a model of ZC + Sigma_2 Replacement, then there is a model of NFU*. Although I don't think I actually invoke the Barwise compactness theorem, ideas very close to that theorem [about infinitary logic and infinitary proofs] play a crucial role in the the imp[rovement from part B to part C. In addition, while the use of L was mainly a convenience in parts A and B, it seems to play an essential role in part C. This ends the high level outline of what we are going to do and with it letter 3. In the next letter, I review term models [essentially these are EM blueprints] and state sufficient conditions on a term model to yield a model of NFU*. From holmes@catseye.idbsu.edu Mon Apr 24 07:50:13 2000 Return-Path: Received: from diamond.idbsu.edu (root@diamond.idbsu.edu [132.178.200.127]) by catseye.idbsu.edu (8.8.7/8.8.7) with ESMTP id HAA07629 for ; Mon, 24 Apr 2000 07:50:13 -0600 Received: from catseye.idbsu.edu (IDENT:holmes@catseye.idbsu.edu [132.178.200.125]) by diamond.idbsu.edu (8.9.3/8.9.3) with ESMTP id HAA07692 for ; Mon, 24 Apr 2000 07:47:33 -0600 From: holmes@catseye.idbsu.edu Received: (from holmes@localhost) by catseye.idbsu.edu (8.8.7/8.8.7) id HAA07624; Mon, 24 Apr 2000 07:50:13 -0600 Date: Mon, 24 Apr 2000 07:50:13 -0600 Message-Id: <200004241350.HAA07624@catseye.idbsu.edu> To: holmes@diamond.idbsu.edu, solovay@math.berkeley.edu Subject: Re: NFU*: Letter1 Status: RO Dear Bob, Thank you for your letter. Yes, we are talking about the same things: your definition of NFU* defines the theory I was asking about, and I understood "ZC + Sigma_2 replacement" correctly. Re your remark: One final remark. I expected the consistency strength of NFU* to be much stronger than it turned out to be. On first glance, it looks rather similar to NFUB. My comment: I thought the same thing! --Randall From solovay@math.berkeley.edu Wed Apr 26 00:21:30 2000 Return-Path: Received: from diamond.idbsu.edu (root@diamond.idbsu.edu [132.178.200.127]) by catseye.idbsu.edu (8.8.7/8.8.7) with ESMTP id AAA11378 for ; Wed, 26 Apr 2000 00:21:30 -0600 Received: from math.berkeley.edu (gold.Math.Berkeley.EDU [169.229.58.61]) by diamond.idbsu.edu (8.9.3/8.9.3) with ESMTP id AAA32145 for ; Wed, 26 Apr 2000 00:18:47 -0600 Received: from blue2.math.Berkeley.EDU (blue2.math.berkeley.edu [169.229.58.60]) by math.berkeley.edu (8.9.3/8.9.3) with ESMTP id XAA25290; Tue, 25 Apr 2000 23:18:43 -0700 (PDT) From: "Robert M. Solovay" Received: (from solovay@localhost) by blue2.math.Berkeley.EDU (8.9.3/8.9.3) id XAA17214; Tue, 25 Apr 2000 23:18:54 -0700 (PDT) Date: Tue, 25 Apr 2000 23:18:54 -0700 (PDT) Message-Id: <200004260618.XAA17214@blue2.math.Berkeley.EDU> To: holmes@diamond.idbsu.edu Subject: NFU*: Description of the term language Cc: solovay@math.berkeley.edu Status: RO We begin phase A of the proof of the converse direction. We are working in the metatheory ZFC + V=L + "There is an inaccessible cardinal theta". We start by defining the concepts of term language and term model. [These concepts are really rather trivial.] Then we list various conditions on a term model and show that if we can meet them we can generate a model of NFU*. Finally, we will give a construction of a term model meeting these requirements. This will complete phase A of the proof. A term language is just a language for first-order logic which has only the predicate =. The following concepts make sense for such a language: term; closed term. A term-model is an equivalence relation on the closed terms so that the resulting structure satisfies the equality axioms. More explicitly, if == is the equivalence relation, f is an n-ary function symbol of the term language, t_1, ..., t_n, and s_1, ..., s_n are closed terms of the language such that for 1 <= i <= n we have s_i == t_i holding in the term model, then f(s_1, ...,s_n) == f(t_1, ...,t_n) holds in the term model. ########################################################## Let us now spell out the particular term language which we will employ in the proof. As I remarked previously, it will depend on the choice of a certain cardinal alpha < theta. The precise value of alpha will be determined in the course of the construction that underlies our proof. Our language will have an infinite stock of variables x_i (for i in omega). For each ordinal gamma < alpha, there will be a corresponding constant gamma_bar. {The intended meaning of gamma_bar is gamma.] For each i in Z [Z is the set of integers, positive, negative or zero] there will be a constant xi_i. The intuition is that the xi_i's are a generating set of indiscernibles. It will turn out that our term-model converts naturally to a model of a set-theory somewhat stronger than KP + MacLane Set Theory plus V= L. In that "model of set-theory" the xi_i's will be lbfp's and the map which sends i to xi_i will be order preserving. [That map will not be a set of the "model of set theory" arising from the term model, of course.] For each positive integer n, and each non-negative integer i, there will be an n-ary function symbol f_{n,i}. We can explain the intended meaning of these by telling what their canonical interpretation is in a model of the form L_lambda [where lambda is a lbfp]. So let lambda be as stated. Let n, i meet the restrictions just given. Let a_1, ..., a_n be elements of L_lambda. We fix a Godel numbering of the formulas of the language of set-theory. [This is the first order language with no constant or function symbols and with just two predicates [both binary] one for = and one for epsilon.] To abbreviate, we write f for f_{n,i}. If i is not the Godel number of a formula of set-theory whose free variables are a subset of {x_0, ..., x_n} then f(a_1, ..., a_n) = 0. Suppose we are not in this case. Let phi(x_0, ...,x_n) be the formula with Godel number i. Let delta be the least bfp such that a_1, ..., a_n are members of L_delta. delta < lambda since lambda is a lbfp. If, in L_delta, there exists an a such that L_delta thinks that phi(a,a_1, ..., a_n), then f(a_1, ..., a_n) is the L-least such a. If there is no such a, then f(a_1, ..., a_n) = 0. Finally, our term language will have for each i in omega, there will be a unary function h_i. The meaning of the h_i's will be decided in the course of our construction. We shall employ this freedom to arrange that in the final model of NFU*. The strongly cantorian ordinals correspond precisely to the ordinals less than alpha. We remark that the functions f_{n,i} play roughly the same role that the local functions that I used in my construction of n-Mahlos in NFUA did. [They form a slightly larger class of functions, however.] This ends letter 4. The next topic to take up is how certain term-models for our language yield models of KP + V=L + MacLane Set Theory + "There are arbitrarily large lbfp's". From solovay@math.berkeley.edu Fri Apr 28 14:28:39 2000 Return-Path: Received: from math.berkeley.edu (gold.Math.Berkeley.EDU [169.229.58.61]) by catseye.idbsu.edu (8.8.7/8.8.7) with ESMTP id OAA02470 for ; Fri, 28 Apr 2000 14:28:38 -0600 Received: from blue2.math.Berkeley.EDU (blue2.math.berkeley.edu [169.229.58.60]) by math.berkeley.edu (8.9.3/8.9.3) with ESMTP id NAA22704; Fri, 28 Apr 2000 13:28:22 -0700 (PDT) From: "Robert M. Solovay" Received: (from solovay@localhost) by blue2.math.Berkeley.EDU (8.9.3/8.9.3) id NAA16306; Fri, 28 Apr 2000 13:28:35 -0700 (PDT) Date: Fri, 28 Apr 2000 13:28:35 -0700 (PDT) Message-Id: <200004282028.NAA16306@blue2.math.Berkeley.EDU> To: holmes@catseye.idbsu.edu Subject: Re: your letters on NFU* Cc: solovay@math.berkeley.edu Status: RO There must be good sources for the finite axiomitizability of Sigma_2 replacement but I don't know one off hand. The basic idea is that there is a universal Sigma_2 relation which has one extra parameter for an integer. Plugging in different integers gets [up to provable equivalence] all Sigma_2 relations. This follows from the corresponding result for Sigma_1. That is in Jensen's magnum opus on the fine structure of L [in section 1 or 2 as I recall]. But that also is easy. It's enough to give a uniform Sigma_1 definition of the truth of Sigma_0 formulas. But this is straightforward using the usual Tarski definition of truth and the fact that a Sigma_0 formula holds in the world iff it holds in a transitive non-empty set containing all its parameters. I don't know that I really use the finite axiomitizability of Sigma_2 replacement. I tend to subconsciously use "everything I know" when constructing a proof. [A coauthor [Volodya Shavrukov has complained about this tendency.] Later, I go through the work of seeing what I really need and how to simplify the proof for presentation. I'm working on the next letter right now. I plan to do the letters that will correspond to phase A of the converse, and then wait for feedback from you before plunging on to the later phases. I'm also doing letter 5 in TeX and will send it to you [via MIME] as a dvi file. Of course, if you have trouble unpacking it, let me know. One way or another, I'll get it to you. --Bob P. S. I should, of course, have a "close bracket" after "Shavrukov". The stupid mail program I use to reply to letters doesn't permit revisions. I should shift to Pine once for all. From solovay@math.berkeley.edu Thu May 4 23:10:16 2000 Return-Path: Received: from diamond.idbsu.edu (root@diamond.idbsu.edu [132.178.200.127]) by catseye.idbsu.edu (8.8.7/8.8.7) with ESMTP id XAA11011 for ; Thu, 4 May 2000 23:10:16 -0600 Received: from math.berkeley.edu (gold-slow.Math.Berkeley.EDU [128.32.183.94]) by diamond.idbsu.edu (8.9.3/8.9.3) with ESMTP id XAA04453 for ; Thu, 4 May 2000 23:09:54 -0600 Received: from blue2.math.Berkeley.EDU (blue2.math.berkeley.edu [169.229.58.60]) by math.berkeley.edu (8.9.3/8.9.3) with ESMTP id WAA18502 for ; Thu, 4 May 2000 22:09:51 -0700 (PDT) Received: from localhost (solovay@localhost) by blue2.math.Berkeley.EDU (8.9.3/8.9.3) with ESMTP id WAA15939 for ; Thu, 4 May 2000 22:10:05 -0700 (PDT) X-Authentication-Warning: blue2.math.berkeley.edu: solovay owned process doing -bs Date: Thu, 4 May 2000 22:10:05 -0700 (PDT) From: "Robert M. Solovay" X-Sender: solovay@blue2 To: Randall Holmes Subject: Letter 5: Making the term model into a model of set-theory Message-ID: MIME-Version: 1.0 Content-Type: MULTIPART/MIXED; BOUNDARY="-559023410-851401618-957503405=:15840" Status: RO This message is in MIME format. The first part should be readable text, while the remaining parts are likely unreadable without MIME-aware tools. Send mail to mime@docserver.cac.washington.edu for more info. ---559023410-851401618-957503405=:15840 Content-Type: TEXT/PLAIN; charset=US-ASCII Randall, Here, finally, is the next installment of the proof of my results concerning NFU*. I hope you have no trouble printing up the attached dvi file. If you do have trouble, let me know, and I can send it as a ps file or [shudder!] as a tex file plus macro files. Part of the reason this took so long to get out, is that I have been experimenting with some TeX macro files that I got from a friend. Occasionally they don't do what I want them to do, and it sometimes takes me a while to wrestle them into behaving. When I constructed the proof in Trondheim, I dismissed the point handled in the following letter as "obvious". It's not quite as obvious as I thought. One thing which perhaps might make the exposition a little easier would be to add a function to the term language which maps a set x to the least bfp beta such that x is in L_\beta. The next letter should (a) observe that the notion of "well-instantiated" guarantees that the xi's are indiscernibles in the term model and (b) give a sufficient condition that the resulting model of NFU is a model of NFU*. --Bob ---559023410-851401618-957503405=:15840 Content-Type: APPLICATION/octet-stream; name="letter5.dvi" Content-Transfer-Encoding: BASE64 Content-ID: Content-Description: Content-Disposition: attachment; filename="letter5.dvi" 9wIBg5LAHDsAAAAAA+gbIFRlWCBvdXRwdXQgMjAwMC4wNS4wNDoyMTUxiwAA AAEAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAD/////jZ/y AACOoAJ7+emNoP2OBheNkgDCuu7zNRryIlYADmZmAAoAAAAGY21ieDEw4Exl dHRlcpEFhRw1jp8YWiqNkVUW/PMyS/FgeQAOZmYACgAAAAVjbXIxMN1UaGWW BMzNdGVybZNtb5pmZ2RlbJNhc5Nhk21vmGRlbJNvZpNzZXSTdGhlb3J5jqkg LjaNkgCcTfTzO4wt+BQADmZmAAoAAAAHY21jc2MxMOZSkJR6b2Jlcpv/Cjx0 lgVwoE0uk1NvbG+QrhR2kf64UmGYeY6kExN8jZIAtWil80aMLfgUAAwAAAAK AAAAB2NtY3NjMTDrRlAulgSIhU8uk0JvkLu7eJM1OTQ5jqGNkgCrj3tFdWdl bmUslgSIhU9Skzk3NDA1jqGNkgCML47zHd/qPHgACgAAAAoAAAAGY210dDEw 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yuban-c.math.berkeley.edu: solovay owned process doing -bs Date: Sat, 13 May 2000 21:16:25 -0700 (PDT) From: "Robert M. Solovay" X-Sender: solovay@blue1 To: Randall Holmes Subject: Letter 6 Message-ID: MIME-Version: 1.0 Content-Type: MULTIPART/MIXED; BOUNDARY="-559023410-851401618-958277785=:13792" Status: RO This message is in MIME format. The first part should be readable text, while the remaining parts are likely unreadable without MIME-aware tools. Send mail to mime@docserver.cac.washington.edu for more info. ---559023410-851401618-958277785=:13792 Content-Type: TEXT/PLAIN; charset=US-ASCII This letter is pretty short and simple. It shows that we can get a model of NFU from the term model in the usual way and gives a sufficient criterion for that model to be a model of NFU*. I think one more letter [somewhat longer] should finish phase A of the converse direction. --Bob ---559023410-851401618-958277785=:13792 Content-Type: APPLICATION/octet-stream; name="letter6.dvi" Content-Transfer-Encoding: BASE64 Content-ID: Content-Description: Content-Disposition: attachment; filename="letter6.dvi" 9wIBg5LAHDsAAAAAA+gbIFRlWCBvdXRwdXQgMjAwMC4wNS4xMzoyMTA3iwAA AAEAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAD/////jZ/y AACOoAJ7+emNoP2OBheNkgDCuu7zNRryIlYADmZmAAoAAAAGY21ieDEw4Exl dHRlcpEFhRw2jp8Y3beNkVZMx/MyS/FgeQAOZmYACgAAAAVjbXIxMN1TdQ5j aWVukJmYdJYEzM1jb25kaXRpb25zk2ZvcpNhk21vkGZnZGVsk29mk/M/GvIi VgAMAAAACgAAAAZjbWJ4MTDqTkZV8z4hIiyaAAwAAAAKAAAABmNtc3kxMOkD jp8hdxaNkgCcTfTzO4wt+BQADmZmAAoAAAAHY21jc2MxMOZSkJR6b2Jlcpv/ Cjx0lgVwoE0uk1NvbG+QrhR2kf64UmGYeY6kE1VCjZIAtWil80aMLfgUAAwA AAAKAAAAB2NtY3NjMTDrRlAulgSIhU8uk0JvkLu7eJM1OTQ5jqGNkgCrj3tF dWdlbmUslgSIhU9Skzk3NDA1jqGNkgCML47zHd/qPHgACgAAAAoAAAAGY210 dDEwyGVtYWlsOpEKf/pzb2xvdmF5QG1hdGguYmVya2VsZXkuZWR1jqGpNFQb 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---559023410-851401618-958277785=:13792-- From solovay@math.berkeley.edu Mon May 15 23:24:16 2000 Return-Path: Received: from diamond.idbsu.edu (root@diamond.idbsu.edu [132.178.200.127]) by catseye.idbsu.edu (8.8.7/8.8.7) with ESMTP id XAA28881 for ; Mon, 15 May 2000 23:24:16 -0600 Received: from math.berkeley.edu (gold.Math.Berkeley.EDU [169.229.58.61]) by diamond.idbsu.edu (8.9.3/8.9.3) with ESMTP id XAA20108 for ; Mon, 15 May 2000 23:23:38 -0600 Received: from yuban-c.math.berkeley.edu (blue1.Math.Berkeley.EDU [169.229.58.58]) by math.berkeley.edu (8.9.3/8.9.3) with ESMTP id WAA19481 for ; Mon, 15 May 2000 22:23:35 -0700 (PDT) Received: from localhost (solovay@localhost) by yuban-c.math.berkeley.edu (8.9.3/8.9.3) with ESMTP id WAA05407 for ; Mon, 15 May 2000 22:23:34 -0700 (PDT) X-Authentication-Warning: yuban-c.math.berkeley.edu: solovay owned process doing -bs Date: Mon, 15 May 2000 22:23:34 -0700 (PDT) From: "Robert M. Solovay" X-Sender: solovay@blue1 To: Randall Holmes Subject: Letter 7: Completion of phase A Message-ID: MIME-Version: 1.0 Content-Type: MULTIPART/MIXED; BOUNDARY="-559023410-851401618-958454614=:5342" Status: RO This message is in MIME format. The first part should be readable text, while the remaining parts are likely unreadable without MIME-aware tools. Send mail to mime@docserver.cac.washington.edu for more info. ---559023410-851401618-958454614=:5342 Content-Type: TEXT/PLAIN; charset=US-ASCII Randall, Here is letter 7. I'm going to wait to hear from you that you've digested this portion of the proof before beginning the description of phase B. Of course, if anything is unclear just let me know and I'll supply further details. I left a few things as "exercises for the reader". It's my hope that you will think about these things a bit before declaring them unclear. Of course, if they remain unclear upon reflection, I'll supply further details. --Bob ---559023410-851401618-958454614=:5342 Content-Type: APPLICATION/octet-stream; name="letter7.dvi" Content-Transfer-Encoding: BASE64 Content-ID: Content-Description: Content-Disposition: attachment; filename="letter7.dvi" 9wIBg5LAHDsAAAAAA+gbIFRlWCBvdXRwdXQgMjAwMC4wNS4xNToyMjE4iwAA AAEAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAD/////jZ/y AACOoAJ7+emNoP2OBheNkgDCuu7zNRryIlYADmZmAAoAAAAGY21ieDEw4Exl dHRlcpEFhRw3jp8WogSNkS4zKfMyS/FgeQAOZmYACgAAAAVjbXIxMN1Db25z dHJ1Y3Rpb26WBMzNb2aTYZNtb5BmZ2RlbJNvZpPzPxryIlYADAAAAAoAAAAG Y21ieDEw6k5GVfM+ISIsmgAMAAAACgAAAAZjbXN5MTDpA91mcm9tk2Fuk2lu YWNjZXNzaWJsZY6pHHc7jZIAnE308zuMLfgUAA5mZgAKAAAAB2NtY3NjMTDm UpCUem9iZXKb/wo8dJYFcKBNLpNTb2xvkK4UdpH+uFJhmHmOpBLMzY2SALVo pfNGjC34FAAMAAAACgAAAAdjbWNzYzEw60ZQLpYEiIVPLpNCb5C7u3iTNTk0 OY6hjZIAq497RXVnZW5lLJYEiIVPUpM5NzQwNY6hjZIAjC+O8x3f6jx4AAoA AAAKAAAABmNtdHQxMMhlbWFpbDqRCn/6c29sb3ZheUBtYXRoLmJlcmtlbGV5 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(blue1.Math.Berkeley.EDU [169.229.58.58]) by math.berkeley.edu (8.9.3/8.9.3) with ESMTP id OAA28884 for ; Tue, 16 May 2000 14:04:01 -0700 (PDT) Received: from localhost (solovay@localhost) by yuban-c.math.berkeley.edu (8.9.3/8.9.3) with ESMTP id OAA12630 for ; Tue, 16 May 2000 14:04:00 -0700 (PDT) X-Authentication-Warning: yuban-c.math.berkeley.edu: solovay owned process doing -bs Date: Tue, 16 May 2000 14:04:00 -0700 (PDT) From: "Robert M. Solovay" X-Sender: solovay@blue1 To: Randall Holmes Subject: Letter 8: Some brief meta-comments Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO Well, next pass through I will make a few small changes: 1) The notion of term-model is misleading. It's really analogous to a theory. 2) The definition of instantiation function is overly sloppy. The problem is there is a slew of languages around [indexed by a) an integer n; b) an ordinal gamma; c) a single bit which is on for "pre" and off if "not-pre". So an instantiation function is a map from the ordinals less than theta to a structure for one of these languages [the same language for all the ordinals less than theta]. Of course, when doing the next pass, I should specify which language is in question at any point. I think the proof is ok; the exposition can certainly be significantly improved. --Bob From holmes@catseye.idbsu.edu Tue May 16 15:06:59 2000 Return-Path: Received: from diamond.idbsu.edu (root@diamond.idbsu.edu [132.178.200.127]) by catseye.idbsu.edu (8.8.7/8.8.7) with ESMTP id PAA30281 for ; Tue, 16 May 2000 15:06:59 -0600 Received: from catseye.idbsu.edu (IDENT:holmes@catseye.idbsu.edu [132.178.200.125]) by diamond.idbsu.edu (8.9.3/8.9.3) with ESMTP id PAA25619 for ; Tue, 16 May 2000 15:06:29 -0600 From: holmes@catseye.idbsu.edu Received: (from holmes@localhost) by catseye.idbsu.edu (8.8.7/8.8.7) id PAA30276; Tue, 16 May 2000 15:06:59 -0600 Date: Tue, 16 May 2000 15:06:59 -0600 Message-Id: <200005162106.PAA30276@catseye.idbsu.edu> To: holmes@diamond.idbsu.edu, solovay@math.berkeley.edu Subject: Re: Letter 8: Some brief meta-comments Status: RO A meta-meta-comment: I have received all letters so far and am trying to read them in parallel with revision of a conference paper; this means I'm reading quite slowly. No problems with printing out letters 5-7. --Randall From solovay@math.berkeley.edu Fri May 19 14:59:06 2000 Return-Path: Received: from math.berkeley.edu (gold-slow.Math.Berkeley.EDU [128.32.183.94]) by catseye.idbsu.edu (8.8.7/8.8.7) with ESMTP id OAA02733 for ; Fri, 19 May 2000 14:59:06 -0600 Received: from blue2.math.Berkeley.EDU (blue2.math.berkeley.edu [169.229.58.60]) by math.berkeley.edu (8.9.3/8.9.3) with ESMTP id NAA16223 for ; Fri, 19 May 2000 13:58:20 -0700 (PDT) Received: from localhost (solovay@localhost) by blue2.math.Berkeley.EDU (8.9.3/8.9.3) with ESMTP id NAA20561 for ; Fri, 19 May 2000 13:58:39 -0700 (PDT) X-Authentication-Warning: blue2.math.berkeley.edu: solovay owned process doing -bs Date: Fri, 19 May 2000 13:58:39 -0700 (PDT) From: "Robert M. Solovay" X-Sender: solovay@blue2 To: holmes@catseye.idbsu.edu Subject: Re: letters 4 and 5 In-Reply-To: <200005192051.OAA02719@catseye.idbsu.edu> Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO Randall, Glad to see my work in writing these up serves some purpose. So far I don't detect any questions I need to answer. If I don't answer a question you want answered, reask FIRMLY. --Bob On Fri, 19 May 2000 holmes@catseye.idbsu.edu wrote: > > Dear Bob, > > The fog is clearing a bit -- I think I'm recovering from the delusion > that I'm on vacation :-) > > At any rate, I now understand and believe letters 4 and 5, though > I think that I will need to peek back under the hood in letter 5. > > Onward to letter 6... > > --Randall > From solovay@math.berkeley.edu Fri May 19 15:06:48 2000 Return-Path: Received: from math.berkeley.edu (gold-slow.Math.Berkeley.EDU [128.32.183.94]) by catseye.idbsu.edu (8.8.7/8.8.7) with ESMTP id PAA02785 for ; Fri, 19 May 2000 15:06:47 -0600 Received: from blue2.math.Berkeley.EDU (blue2.math.berkeley.edu [169.229.58.60]) by math.berkeley.edu (8.9.3/8.9.3) with ESMTP id OAA16640 for ; Fri, 19 May 2000 14:06:02 -0700 (PDT) Received: from localhost (solovay@localhost) by blue2.math.Berkeley.EDU (8.9.3/8.9.3) with ESMTP id OAA20591 for ; Fri, 19 May 2000 14:06:21 -0700 (PDT) X-Authentication-Warning: blue2.math.berkeley.edu: solovay owned process doing -bs Date: Fri, 19 May 2000 14:06:21 -0700 (PDT) From: "Robert M. Solovay" X-Sender: solovay@blue2 To: holmes@catseye.idbsu.edu Subject: Re: a suggestion In-Reply-To: <200005192100.PAA02736@catseye.idbsu.edu> Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO Randall, It's certainly fine to suggest it. I'll probably say no because (a) I have a paper on quantum computation which I **must** write up and get out and (b) I generally prefer to publish in journals rather than special volumes. Although my track record in recent years is not very good about getting papers out, I do intend to write up my results on variants of NFU for publication. But first (a) I want to write up my results on "Lie Groups and Quantum Circuits" [the paper alluded to above] and (b) my work on Gleason's theorem for non-separable Hilbert spaces --Bob On Fri, 19 May 2000 holmes@catseye.idbsu.edu wrote: > > Dear Bob, > > Marcel Crabbe (crabbe@risp.ucl.ac.be) recently invited me to contribute > to a volume in honor of Maurice Boffa's 60th birthday. It occurred to me that > some of the stuff on consistency strength of extensions of NFU that you have > shared with me recently might be appropriate for this venue, so I suggested > to Marcel that he ask you if you had something to contribute. > > Of course, your state of retirement gives you the privilege of ignoring > all such suggestions :-) > > --Randall > From solovay@math.berkeley.edu Mon Jan 1 19:45:34 2001 Return-Path: Received: from diamond.boisestate.edu (IDENT:root@diamond.boisestate.edu [132.178.208.127]) by catseye.idbsu.edu (8.9.3/8.9.3) with ESMTP id TAA05070 for ; Mon, 1 Jan 2001 19:45:24 -0700 Received: from math.berkeley.edu (gold.Math.Berkeley.EDU [169.229.58.61]) by diamond.boisestate.edu (8.9.3/8.9.3) with ESMTP id TAA27130 for ; Mon, 1 Jan 2001 19:36:42 -0700 Received: from blue2.math.Berkeley.EDU (blue2.math.berkeley.edu [169.229.58.60]) by math.berkeley.edu (8.9.3/8.9.3) with ESMTP id SAA00058 for ; Mon, 1 Jan 2001 18:35:48 -0800 (PST) Received: from localhost (solovay@localhost) by blue2.math.Berkeley.EDU (8.9.3/8.9.3) with ESMTP id SAA13143 for ; Mon, 1 Jan 2001 18:35:48 -0800 (PST) X-Authentication-Warning: blue2.math.berkeley.edu: solovay owned process doing -bs Date: Mon, 1 Jan 2001 18:35:48 -0800 (PST) From: "Robert M. Solovay" X-Sender: solovay@blue2 To: Randall Holmes Subject: Apologies Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO Randall, It's taking me longer than expected to get back to you re the stuff you asked about. I'm *still* writing up that paper I mentioned. And I've taken a consulting gig. I do intend to get back to you; it's just hard to say precisely when. You have my permission to gently nag me every three months or so, if I haven't yet satisfied your curiosity. --Bob From solovay@yahoo.com Wed Feb 21 15:26:54 2001 Return-Path: Received: from web1611.mail.yahoo.com (web1611.mail.yahoo.com [128.11.23.177]) by catseye.idbsu.edu (8.9.3/8.9.3) with SMTP id PAA01474 for ; Wed, 21 Feb 2001 15:26:53 -0700 Received: (qmail 27809 invoked by uid 60001); 21 Feb 2001 22:25:34 -0000 Message-ID: <20010221222534.27808.qmail@web1611.mail.yahoo.com> Received: from [192.203.205.129] by web1611.mail.yahoo.com; Wed, 21 Feb 2001 14:25:34 PST Date: Wed, 21 Feb 2001 14:25:34 -0800 (PST) From: Robert Solovay Subject: Request for help To: holmes@catseye.idbsu.edu In-Reply-To: <200004172129.PAA31311@catseye.idbsu.edu> MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Status: RO Randall, I plan to start thinking again about NFU*. [I may give a logic colloquium talk on these results at UCLA.] Unfortunately, I'm on the road at the moment, so I have no access to my letters to you re all this. If you've kept copies and could send them to me it would be very helpful. Thanks in advance. --Bob __________________________________________________ Do You Yahoo!? Yahoo! Auctions - Buy the things you want at great prices! http://auctions.yahoo.com/ From solovay@ccrwest.org Mon Feb 26 12:40:05 2001 Return-Path: Received: from diamond.boisestate.edu (IDENT:root@diamond.boisestate.edu [132.178.208.127]) by catseye.idbsu.edu (8.9.3/8.9.3) with ESMTP id MAA06155 for ; Mon, 26 Feb 2001 12:40:05 -0700 Received: from bluenote.ccrwest.org (smap@bluenote.ccrwest.org [192.203.205.129]) by diamond.boisestate.edu (8.9.3/8.9.3) with SMTP id MAA12574 for ; Mon, 26 Feb 2001 12:38:54 -0700 Received: by bluenote.ccrwest.org (4.1/CCRWEST-I1.20) id AA07881; Mon, 26 Feb 2001 11:38:27 PST Received: from ccrwest.ccrwest.org(192.203.205.65) by bluenote.ccrwest.org via smap (V1.3) id sma007879; Mon, 26 Feb 2001 11:38:22 -0800 Received: from grazia.ccrwest.org by ccrwest.ccrwest.org (4.1/CCRWEST-2.13) id AA05916; Mon, 26 Feb 2001 11:38:22 PST Received: from localhost by grazia.ccrwest.org (4.1/ccrwest-1.6) id AA01717; Mon, 26 Feb 2001 11:38:21 PST Date: Mon, 26 Feb 2001 11:38:20 -0800 (PST) From: "Robert M. Solovay" To: "M. Randall Holmes" Subject: Re: Remarks on your retraction In-Reply-To: <200102261607.JAA05985@catseye.idbsu.edu> Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO Randall, Yes V=L is eliminable from the statement calculating the consistency strength of "NFU + Counting" [getting an equivalent true characterization]. --Bob From solovay@ccrwest.org Mon Feb 26 12:59:04 2001 Return-Path: Received: from diamond.boisestate.edu (IDENT:root@diamond.boisestate.edu [132.178.208.127]) by catseye.idbsu.edu (8.9.3/8.9.3) with ESMTP id MAA06162 for ; Mon, 26 Feb 2001 12:59:04 -0700 Received: from bluenote.ccrwest.org (smap@bluenote.ccrwest.org [192.203.205.129]) by diamond.boisestate.edu (8.9.3/8.9.3) with SMTP id MAA13037 for ; Mon, 26 Feb 2001 12:57:48 -0700 Received: by bluenote.ccrwest.org (4.1/CCRWEST-I1.20) id AA08179; Mon, 26 Feb 2001 11:57:28 PST Received: from ccrwest.ccrwest.org(192.203.205.65) by bluenote.ccrwest.org via smap (V1.3) id sma008169; Mon, 26 Feb 2001 11:57:18 -0800 Received: from grazia.ccrwest.org by ccrwest.ccrwest.org (4.1/CCRWEST-2.13) id AA06074; Mon, 26 Feb 2001 11:57:04 PST Received: from localhost by grazia.ccrwest.org (4.1/ccrwest-1.6) id AA01811; Mon, 26 Feb 2001 11:57:02 PST Date: Mon, 26 Feb 2001 11:57:01 -0800 (PST) From: "Robert M. Solovay" To: holmes@diamond.boisestate.edu Subject: Re: Remarks on your retraction (fwd) Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO ---------- Forwarded message ---------- Date: Mon, 26 Feb 2001 09:07:38 -0700 From: "M. Randall Holmes" To: holmes@diamond.boisestate.edu, solovay@ccrwest.org Subject: Re: Remarks on your retraction Dear Bob, You are indeed correct. I think that I am guilty in this instance of having thought one thing and written another (though I am certainly capable of thinking absurd things, alas). Details follow. --Randall \begin{quote} The claims "A set is strongly Cantorian if all of its subsets are Cantorian" and "A cardinal is strongly Cantorian if all smaller cardinals are strongly Cantorian" is false. The paper itself shows that there is a counterexample, if one reads it carefully! I produce a model there in which AxCount holds and $\aleph_{\aleph_{\omega}}$ does not exist. In this model, $\aleph_{\omega}$ must be Cantorian but not strongly Cantorian. Now suppose in addition that the GCH holds in the model (as is easily arranged, as for example by assuming V=L in the underlying set theory in which the model is constructed). It follows that every subset of a set of size \aleph{omega} is cantorian (because each subset is either of size $\aleph{\omega}$ or of one of the sizes $n$ (a natural number) or $\aleph_n$, which are all cantorian if AxCount holds). \end{quote} Dear Bob, You say: But the following results which contradict the second claim are easy to establish: A set is strongly cantorian iff its cardinal is s. c. iff one [equivalently all] ordinals of that cardinality are s. c. A cardinal kappa is s.c. iff every cardinal lambda < kappa is s. c. Thus the second claim is **true** I reply: You are right, of course. The counterexample I give (though I describe it correctly) is not a counterexample to the second claim (and I'm not sure I meant to claim that it was -- I may not have written the sentence I was actually meaning to write -- see below). Some nonstandard $aleph_n$ will fail to be s.c. in the model described (and I _knew_ that!) If a cardinal is s.c. it is obvious that all subsets of a set with that cardinality are s.c., so all cardinals less than that cardinal are s.c. Now suppose that all cardinals less than a cardinal kappa are s.c., and that kappa itself is not s.c. Then kappa (here identified with its initial ordinal) would be the smallest non-s.c. ordinal, which is absurd. That is indeed easy to prove. I am really wondering why I said that. The first claim was of course the issue, and what was startling me was that (as the falsehood of the first claim shows) the non-s.c. character of a cardinal (understood as an initial ordinal) must be witnessed by the non-s.c. character of a smaller ordinal but need not be witnessed by the s.c. character of a smaller cardinal. I remember thinking about this, and I do not remember thinking the absurd thing I actually wrote (and, as I remarked above, I knew perfectly well that the aleph_n's in the model I describe above are not all s.c. -- I only assert that they are all cantorian in the quoted text). It would have been correct and relevant to give as the second plausible but false sentence (*) A cardinal is s.c. iff each smaller cardinal is cantorian and I think that was actually what I meant to write. You wrote: While thinking about all this I noticed the following which is perhaps new; The consistency strength of NFU + Axiom of Counting is precisely the same as the following: (A) ZF- (B) V = L (C) The following scheme of axioms: For each standard integer n, there is an axiom that asserts that aleph_{aleph_n} exists. Corrollary: Assume that NFU + Axiom of Counting is consistent. Then it has a model in which aleph_{aleph_n} does not exist for some [necessarily non-standard] natural number n. I comment: This does look new, though I think you said something similar in an earlier note. There was a "folklore" claim about the precise strength of NFU + AxCount circulating while I was in Belgium in 1990-1, but I forget what it was! I assume the clause V=L is actually eliminable? I already knew the corollary: the model I describe in the "strong axioms" paper (and allude to in the quoted text) has this property. From solovay@ccrwest.org Sat Feb 24 01:06:15 2001 Return-Path: Received: from diamond.boisestate.edu (IDENT:root@diamond.boisestate.edu [132.178.208.127]) by catseye.idbsu.edu (8.9.3/8.9.3) with ESMTP id BAA03628 for ; Sat, 24 Feb 2001 01:06:15 -0700 Received: from bluenote.ccrwest.org (smap@bluenote.ccrwest.org [192.203.205.129]) by diamond.boisestate.edu (8.9.3/8.9.3) with SMTP id BAA10909 for ; Sat, 24 Feb 2001 01:04:50 -0700 Received: by bluenote.ccrwest.org (4.1/CCRWEST-I1.20) id AA22375; Sat, 24 Feb 2001 00:04:48 PST Received: from ccrwest.ccrwest.org(192.203.205.65) by bluenote.ccrwest.org via smap (V1.3) id sma022373; Sat, 24 Feb 2001 00:04:08 -0800 Received: from makam.ccrwest.org by ccrwest.ccrwest.org (4.1/CCRWEST-2.13) id AA01410; Sat, 24 Feb 2001 00:04:08 PST Received: from localhost by makam.ccrwest.org (4.1/ccrwest-1.6) id AA27436; Sat, 24 Feb 2001 00:04:06 PST Date: Sat, 24 Feb 2001 00:04:06 -0800 (PST) From: "Robert M. Solovay" To: holmes@diamond.boisestate.edu Subject: Remarks on your retraction Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO Dear Randall, As part of the process of thinking my way into things NFish again [so that I may reconstruct the missing parts of the proofs of my results re NFU*] I reread your retraction letter from April 2000. There are comments made there that now seem dubious to me. You write: \begin{quote} The claims "A set is strongly Cantorian if all of its subsets are Cantorian" and "A cardinal is strongly Cantorian if all smaller cardinals are strongly Cantorian" is false. The paper itself shows that there is a counterexample, if one reads it carefully! I produce a model there in which AxCount holds and $\aleph_{\aleph_{\omega}}$ does not exist. In this model, $\aleph_{\omega}$ must be Cantorian but not strongly Cantorian. Now suppose in addition that the GCH holds in the model (as is easily arranged, as for example by assuming V=L in the underlying set theory in which the model is constructed). It follows that every subset of a set of size \aleph{omega} is cantorian (because each subset is either of size $\aleph{\omega}$ or of one of the sizes $n$ (a natural number) or $\aleph_n$, which are all cantorian if AxCount holds). \end{quote} But the following results which contradict the second claim are easy to establish: A set is strongly cantorian iff its cardinal is s. c. iff one [equivalently all] ordinals of that cardinality are s. c. A cardinal kappa is s.c. iff every cardinal lambda < kappa is s. c. Thus the second claim is **true**. While thinking about all this I noticed the following which is perhaps new; The consistency strength of NFU + Axiom of Counting is precisely the same as the following: (A) ZF- (B) V = L (C) The following scheme of axioms: For each standard integer n, there is an axiom that asserts that aleph_{aleph_n} exists. Corrollary: Assume that NFU + Axiom of Counting is consistent. Then it has a model in which aleph_{aleph_n} does not exist for some [necessarily non-standard] natural number n. As ever, Bob From solovay@yahoo.com Sat Mar 3 10:46:03 2001 Return-Path: Received: from diamond.boisestate.edu (IDENT:root@diamond.boisestate.edu [132.178.208.127]) by catseye.idbsu.edu (8.9.3/8.9.3) with ESMTP id KAA02122 for ; Sat, 3 Mar 2001 10:46:02 -0700 Received: from web1601.mail.yahoo.com (web1601.mail.yahoo.com [128.11.23.201]) by diamond.boisestate.edu (8.9.3/8.9.3) with SMTP id KAA26250 for ; Sat, 3 Mar 2001 10:44:22 -0700 Received: (qmail 11145 invoked by uid 60001); 3 Mar 2001 17:44:16 -0000 Message-ID: <20010303174416.11144.qmail@web1601.mail.yahoo.com> Received: from [152.163.213.74] by web1601.mail.yahoo.com; Sat, 03 Mar 2001 09:44:16 PST Date: Sat, 3 Mar 2001 09:44:16 -0800 (PST) From: Robert Solovay Subject: RETRACTION re NFU* To: "M. Randall Holmes" Cc: solovay@math.berkeley.edu In-Reply-To: <200102221503.IAA02008@catseye.idbsu.edu> MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Status: RO Randall, I've been thinking about NFU* the last couple of days. I am now sceptical that I ever had a proof of the equiconistency of NFU* and [ZC + Sigma_2 replacement]. The following results still look correct to me: 1) If NFU* is consistent, then so is ZC + Sigma_2-replacement. 2) If ZC + Sigma_3-replacement is consistent, then so is NFU*. I have been through these proofs again in the last couple of days. However my focus has been getting the missing equiconsistency result rather than certifying 1) and 2). My opinion as to whether I can figure out what's going on fluctuates wildly. At any rate, I thought rather hard about all this yesterday, and after some maintenance chores [paying bills, etc.] I intend to take up the cudgles again today. I'll keep you informed. Briefly, the problem is this. One has a carefully prepared model, N, of ZC + Sigma_n-replacement, [n might be 2 or 3]. In the proof that works [for 3], the model of NFU one constructs is a class of N, and this is crucial in verifying the key nex axiom of NFU*. In my proposed approach to the case n=2, one is working in forcing extensions of N; but then the proof of the key new axiom breaks down. Needless to say, I feel somewhat chagrinned about all this. --Bob __________________________________________________ Do You Yahoo!? Get email at your own domain with Yahoo! Mail. http://personal.mail.yahoo.com/ From solovay@ccrwest.org Mon Mar 5 14:27:32 2001 Return-Path: Received: from diamond.boisestate.edu (IDENT:root@diamond.boisestate.edu [132.178.208.127]) by catseye.idbsu.edu (8.9.3/8.9.3) with ESMTP id OAA04250 for ; Mon, 5 Mar 2001 14:27:32 -0700 Received: from bluenote.ccrwest.org (smap@bluenote.ccrwest.org [192.203.205.129]) by diamond.boisestate.edu (8.9.3/8.9.3) with SMTP id OAA10116 for ; Mon, 5 Mar 2001 14:25:44 -0700 Received: by bluenote.ccrwest.org (4.1/CCRWEST-I1.20) id AA18926; Mon, 5 Mar 2001 13:25:22 PST Received: from ccrwest.ccrwest.org(192.203.205.65) by bluenote.ccrwest.org via smap (V1.3) id sma018921; Mon, 5 Mar 2001 13:24:37 -0800 Received: from opus.ccrwest.org by ccrwest.ccrwest.org (4.1/CCRWEST-2.13) id AA05293; Mon, 5 Mar 2001 13:24:37 PST Received: from localhost by opus.ccrwest.org (4.1/ccrwest-1.6) id AA13140; Mon, 5 Mar 2001 13:24:35 PST Date: Mon, 5 Mar 2001 13:24:35 -0800 (PST) From: "Robert M. Solovay" To: holmes@diamond.boisestate.edu Subject: The proof is [perhaps?] up again. Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO Randall, I once again believe that I can show that the consistency strength of NFU* is precisely that of ZC + Sigma_2 replacement. The proof I have now is definitely not the one I thought I had a year ago. Amusingly enough, it also involves the use of infinitary logic in a crucial way. I'm off to lunch. Then I'm going to start writing the proof down in a series of letters to you. As i had previously planned, I will start with the proof of Con(NFU*) from Con(ZC + Sigma_3-replacement). That proof is fairly straightforward, and modulo stuff I need for the Sigma_2-proof is fairly short. On the other hand, the Sigma_2 proof is somewhat subtle and involves substantial new ideas. A lot of what I do is implicit in Barwise, but I have a rather different perspective on things than he does. Unfortunately, though I owe a substantial intellectual debt to Barwise, I can't just quote his results but have to prove analogues for the context in which I work. Since the proof is rather tricky [I'm sure in two weeks I'll be dismissing it as straightforward] I can't be sure that it's correct until I write it up. So if you wish to just save the letters and watch with bemused detachment until the whole series of letters is complete and no bug has emerged that's fine with me. --Bob From solovay@ccrwest.org Mon Mar 5 17:36:30 2001 Return-Path: Received: from diamond.boisestate.edu (IDENT:root@diamond.boisestate.edu [132.178.208.127]) by catseye.idbsu.edu (8.9.3/8.9.3) with ESMTP id RAA04389 for ; Mon, 5 Mar 2001 17:36:30 -0700 Received: from bluenote.ccrwest.org (smap@bluenote.ccrwest.org [192.203.205.129]) by diamond.boisestate.edu (8.9.3/8.9.3) with SMTP id RAA13172 for ; Mon, 5 Mar 2001 17:34:41 -0700 Received: by bluenote.ccrwest.org (4.1/CCRWEST-I1.20) id AA21662; Mon, 5 Mar 2001 16:34:30 PST Received: from ccrwest.ccrwest.org(192.203.205.65) by bluenote.ccrwest.org via smap (V1.3) id sma021658; Mon, 5 Mar 2001 16:34:26 -0800 Received: from grazia.ccrwest.org by ccrwest.ccrwest.org (4.1/CCRWEST-2.13) id AA06067; Mon, 5 Mar 2001 16:34:26 PST Received: from localhost by grazia.ccrwest.org (4.1/ccrwest-1.6) id AA09143; Mon, 5 Mar 2001 16:34:25 PST Date: Mon, 5 Mar 2001 16:34:24 -0800 (PST) From: "Robert M. Solovay" To: Randall Holmes Subject: Phase B:letter 1 Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO Randall, Here starts my presentation of phase B. As I mentioned earlier, the proof of phase C [which is the final desired result] builds on the work of phase B. We are working in the metatheory ZF- [or what comes to the same thing, second order number theory]. It would be fairly routine to carry out the proof in Peano Arithmetic. Probably, the proof could be carried out in PRA as well, but I certainly have not thought through the necessary details. We are given a model of ZC + Sigma_3 Replacement. Our task is to produce a model of NFU*. The first order of business is to massage the model to one with certain extra properties. That task [and some preliminary comments about what one can do in the theory ZC + Sigma_n-replacement + V=L] will be the contents of the present letter. Let's call the given model N_0. We apply Godel's construction inside it getting a model of the theory "ZC + Sigma_3-replacement + V=L". We use T_n as an abbreviation for "ZC + V=L + Sigma_n-replacement". We let N_1 be the model of T_3 just constructed. Work inside N_1. If there is a limit cardinal lambda such that L_lambda is a model of T_2, choose the least such and let L_lambda be the model N_2. If not, let N_2 be N_1. I am being slightly sloppy here. Really, if the first case arises, N_2 is the model whose underlying set is the set of all elements of the underlying set of N_1 that N_1 thinks are in L_lambda. [And the epsilon relation of N_2 is the restriction of the epsilon relation of N_1.] In what follows, we refer to N_2 as N. It is a model of T_3 + "For no limit cardinal lambda is L_lambda a model of T_2". Caution: There is absolutely no reason to think that the ordinals of N are well-ordered [or even that all the integers of N are standard]. What we have gained by this manouvering is the following property: Lemma 1. N thinks: There is a definable map of omega cofinally into the ordinals. Remarks: 1) One can actually show that the map in question is Sigma_4. We won't have need of this more precise result 2) An entirely analogous result holds for T_2: If "ZC + Sigma_2-replacement" is consistent, then there is a model N of "T_2 + 'for no limit cardinal lambda is L_lambda a model of T_2' ". In this model N, there is a definable map from omega cofinally into the ordinals. We shall have need of this analogous result when we tackle phase C. Before beginning the proof of Lemma 1, we state the following proposition whose proof will be left as an exercise to the reader. Proposition: Let n be a positive integer. Then the following are theorem schemes of T_n: A) If R(x,y) is a Sigma_n-relation, then R can be uniformized. That is there is a Sigma_n-relation S such that: (a) S(x,y) implies R(x,y). (b) If R(x,y), then for some z, S(x,z). (c) If S(x,z_1) and S(x,z_2), then z_1 = z_2. B) Suppose S(x) is a Sigma_n formula. [S might have free variables other than x that we are not displaying.] Then the formula (for all x in y) S(x) is equivalent to a Sigma_n formula. [We require, of course, that the variables x and y are distinct.] C) Definition of functions by transfinite induction: Let F be a Sigma_n class which is functional. I.e., if F(x,y) and F(x,z) then y = z. We are concerned with functions G defined on initial segments of OR [possibly all of OR} that satisfy the following equation: G(alpha) = F(G restricted to alpha). [for every alpha in the domain of G] Then either (a) for every ordinal alpha there is such a G with domain alpha or (b) there is a least ordinal gamma such that no such G exists with domain gamma. In the latter case, gamma must be a successor ordinal [say beta + 1] and F is not defined on the function on beta [call it g_beta] that satisfies the above functional equation. In case (a), the various g_beta's piece together to give a class function mapping OR to V, say G, which satisfies the functional equation. This G is Sigma_n. [Or if there is a maximal set-sized g, it is Sigma_n in the same parameters as F.] Some comments on the proof of A through C: These are closely analogous to results of Jensen about transitive models of V=L. The fact that we have full selection is a great help in the proofs. The proofs proceed by induction on n. For a fixed n, one proceeds in the order A, B, C in giving the proofs. We return to the proof of Lemma 1. First using C2, we see that T_2 proves that for every ordinal alpha there is a lbfp [limit of Beth fixed points] greater than alpha. {One first shows that for every alpha, Beth_alpha exists; then one shows that there are arbitrarily large Beth fixed points. Finally, one proves the stated result about lbfp's. Next, as a piece of temporary notation, say that an ordinal beta is n-good for alpha [alpha is also an ordinal] if: 1) beta is greater than alpha; 2) beta is a lbfp; 3) Every Sigma_n formula which is true of some x and which has parameters in L_alpha is true of some x in L_beta. Using full selection plus Sigma_3-uniformization and replacement, one sees, in T_3, that for every alpha there is an ordinal beta which is 3-good for alpha. Next using the fact that we have full induction on omega, one defines a class map H from omega into OR as follows: H(0) = omega. H(n+1) is the least ordinal which is 3-good for H(n). If the range of H was not cofinal in OR, we could define kappa = sup range H. [This uses full selection since I have not estimated the logical complexity of H.] But then L_kappa would easily be seen to be a model of T_3, contrary to our choice of the model N. This completes the proof of Lemma 1 and with it this letter. From solovay@ccrwest.org Mon Mar 5 23:32:33 2001 Return-Path: Received: from diamond.boisestate.edu (IDENT:root@diamond.boisestate.edu [132.178.208.127]) by catseye.idbsu.edu (8.9.3/8.9.3) with ESMTP id XAA04505 for ; Mon, 5 Mar 2001 23:32:33 -0700 Received: from bluenote.ccrwest.org (smap@bluenote.ccrwest.org [192.203.205.129]) by diamond.boisestate.edu (8.9.3/8.9.3) with SMTP id XAA14920 for ; Mon, 5 Mar 2001 23:30:48 -0700 Received: by bluenote.ccrwest.org (4.1/CCRWEST-I1.20) id AA24109; Mon, 5 Mar 2001 22:30:46 PST Received: from ccrwest.ccrwest.org(192.203.205.65) by bluenote.ccrwest.org via smap (V1.3) id sma024101; Mon, 5 Mar 2001 22:30:14 -0800 Received: from opus.ccrwest.org by ccrwest.ccrwest.org (4.1/CCRWEST-2.13) id AA07215; Mon, 5 Mar 2001 22:30:14 PST Received: from localhost by opus.ccrwest.org (4.1/ccrwest-1.6) id AA13888; Mon, 5 Mar 2001 22:30:12 PST Date: Mon, 5 Mar 2001 22:30:12 -0800 (PST) From: "Robert M. Solovay" To: Randall Holmes Subject: Letter 2 of phase B. Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO Randall, This letter covers a small but important point that could have appeared in letter 1. Let me start with a fact that I won't prove [or use in the proof of phase B.] It explains why the naive proof requires T_3 rather than T_2. FACT: Work in T_2. Suppose that for no limit cardinal lambda is L_lambda a model of T_2. Then there is a Sigma_2 map F from OR into omega such that for any integer n, the class of alpha such that F(alpha) = n is bounded in OR. [I.e., it is a *set*.] Lemma 3. [T_3]. Let F be a Sigma_2 map from OR into some ordinal gamma, Then for some ordinal eta less than gamma, the class of alpha such tat F(alpha) = eta is cofinal in OR Remark: Of course, there is no contradiction between Lemma 3 and the FACT. Together, they do establish that T_3 proves that there is a limit ordinal lambda such that L_lambda is a model of T_2. Proof of Lemma 3: We work in T_3, and assume that F counterinstances the lemma. We shall derive a contradiction. Define a relation R(eta, theta) as follows: R(eta, theta) holds if theta is an ordinal, eta is an ordinal less than gamma, and for all ordinals delta >= theta F(delta) is unequal to eta. Claim 1: R is Pi_2: Proof: R can be written: 1) eta is an ordinal; 2) theta is an ordinal; 3) eta is a member of gamma; 4) For all delta, xi: If delta is an ordinal and xi is an ordinal and F(delta) = xi, then xi is unequal to eta. Clauses 1 through 3 are Delta_0. It suffices to see that the last two lines of clause 4) are Pi_2. But these lines are an implication whose hypothesis is Sigma_2 and whose conclusion is Delta_0, so this is clear. Our assumptions that F counterinstances the lemma imply that for every eta < gamma, there is a theta such that R(eta, theta). Uniformize R by a Sigma_3 function H mapping gamma into OR. By Sigma_3 replacement, there is an ordinal xi which is greater than every element of range H. But then there is no possible value for F(xi). [If F(xi) = delta, then this contradicts the facts that: (a) xi > H(delta); hence (b) R(delta, H(delta)), so (c) F(xi) is unequal to delta. ] This completes the proof of the lemma and so ends letter 2 of phase B. From solovay@ccrwest.org Tue Mar 6 14:54:17 2001 Return-Path: Received: from diamond.boisestate.edu (IDENT:root@diamond.boisestate.edu [132.178.208.127]) by catseye.idbsu.edu (8.9.3/8.9.3) with ESMTP id OAA05510 for ; Tue, 6 Mar 2001 14:54:17 -0700 Received: from bluenote.ccrwest.org (smap@bluenote.ccrwest.org [192.203.205.129]) by diamond.boisestate.edu (8.9.3/8.9.3) with SMTP id OAA25787 for ; Tue, 6 Mar 2001 14:52:31 -0700 Received: by bluenote.ccrwest.org (4.1/CCRWEST-I1.20) id AA01509; Tue, 6 Mar 2001 13:52:14 PST Received: from ccrwest.ccrwest.org(192.203.205.65) by bluenote.ccrwest.org via smap (V1.3) id sma001500; Tue, 6 Mar 2001 13:51:38 -0800 Received: from opus.ccrwest.org by ccrwest.ccrwest.org (4.1/CCRWEST-2.13) id AA11571; Tue, 6 Mar 2001 13:51:38 PST Received: from localhost by opus.ccrwest.org (4.1/ccrwest-1.6) id AA14857; Tue, 6 Mar 2001 13:51:37 PST Date: Tue, 6 Mar 2001 13:51:36 -0800 (PST) From: "Robert M. Solovay" To: Randall Holmes Subject: Letter 3 of Phase B Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO This completes my discussion of Phase B of the proof. Comments and questions welcomed. I plan to start my discussion of phase C after lunch. 1. Let's start this letter by reviewing the construction in phase A: 1) We introduced a certain term language L. 2) By a length omega construction, we constructed a term model M. 3) From the model M, we easily derived a model of set-theory with an automorphism. I think I used the same notation for this model as for the term-model. But now I will call it M*. 4) From M*, we derived in the usual way a model Q of NFU. Steps we had taken in the construction of M ensured that Q was in fact a model of NFU*. Recall from Letter 1, that we carefully prepared a model N of T_2. The first key difference in our current construction will be that everything is internal to N. Thus the language L will be described by a certain class of N. And the models M, M*, and Q will be proper classes of N. Also, note that our construction in phase A was indexed by the set omega of non-negative integers. Since our current construction will be internal to N, it will be indexed by the integers of N. In phase A, our term language was determined by a certain ordinal alpha. We didn't specify alpha in advance, but let it be dynamically determined during the course of the construction. What plays the role of alpha in our current construction is the class of all ordinals of the model N. With this as a guide, it should be clear what the terms of our term language are. Simply replace in the old definition the phrase "ordinal less than alpha" by the word "ordinal" and carry out the definition inside N. 2. We pause to indicate how the verification of the crucial axiom of NFU* will be verified. [Selection from s. c. sets using arbitrary formulas.] Work in NFU for the moment. There is the model Z consisting of isomorphism classes of topped well-founded extensional structures. As a class of Z, we have W consisting of those elements of Z whose transitive closures have strongly Cantorian cardinalities. We shall arrange the construction of Q, so that the analogue of W is canonically isomorphic to our starting model N of T_3. Since Q is coded by a class of N, any "definable in Q" subcollection of a strongly Cantorian ordinal, will be "definable in N". Hence it will lie in N, and be constructible at some ordinal stage lying in the strongly Cantorian part of Q. So it will lie in Q. 3. One of the things achieved in letter 1 was that there was a definable map of omega into the ordinals whose range was cofinal with OR. It is easy to massage this map into one with the stated property whose range consists of lbfp's and which is strictly increasing. We let the map be n --> gamma_n. Our construction of our final term-model will closely approximate the one done in phase A. In particular, alpha will be obtained as the limit of a strictly increasing sequence of smaller ordinals alpha_n. To achieve that the limit of the alpha_n's is indeed the class of all ordinals of N, we shall require that alpha_n > gamma_n. Recall that in phase A, a key role was played by instantiation functions. In our current context, the corresponding functions will all be proper classes of N. We will deal with this difficulty as follows: All the instantiation functions we use will be Sigma_2. We will "Godel number" the Sigma_2 functions in a moment in a straightforward way. The Godel-numbers will be certain sets in N. The construction will refer to an instantiation function by giving its Godel number. So how do we Godel number Sigma_2 functions: By pairs . Here i is an integer which is the Godel number of A Sigma_2 formula with three free variables, say x, y, z. We uniformize phi_i in the variable y [getting a map from V^2 to V] which need not be total. We plug in p for the variable z, getting the partial function from V to V with Godel number . 4. Our construction of the model M, will be by defining a function f from omega to V^3 [all this is done inside N, of course]. The function f will be a proper class of N, and definitely not be given by a set of N. Suppose that f(n) = . Then these will have the following significance; a_n will be an ordinal. In fact a_n is precisely what we have previously referred to as alpha_n. We will arrange that the sequence alpha_n is strictly increasing in n, that alpha_n is an lbfp, and that alpha_n > gamma_n. b_n is an equivalence relation on the terms of rank at most (alpha_n,n). We require of course that b_n allows the various relevant functions of our language that are available at stage n to be well-defined on equivalence classes. c_n is the Godel number of a Sigma_2 function with domain the cardinals of N which shows that the equivalence relation b_n is "very-well-instantiated". Logically, to describe the construction, I should specify f(0) and then show how to obtain f(n+1) from f(n). But I shall leave the specification of f(0) "to the reader" and turn to the specification of f(n+1) from f(n). This follow closely the material in letter7.dvi that I sent you in my discussion of part A of the proof, and I shall refer to that manuscript. I shall assume that n > 0. If n = 0, we make no attempt to control the strongly Cantorian ordinals, and the passage from n to n+1 is slightly easier. Following section 2.2 of letter 7, we define (a) the cantorian terms of rank (alpha_{n-1}, n-1); (b) The set W_n of equivalence classes of such terms that are destined to denote ordinals, and (c) the set W_n^\star of equivalence classes of such terms. It is again clear that W_n^\star is well-ordered [in the way described in section 2.2]. We can define the notion of a divergent equivalence class as before. To refine an instantiation function to witness divergence merely uses Sigma_2 uniformization. We can again introduce h_n to insure that the divergent terms will not give strongly Cantorian ordinals. We now start using the fact that we have Sigma_3 replacement. Some of these uses are essential [as is shown by the FACT in letter 2 of phase B]. I haven't checked that they all are. The relation that gamma is an upper bound for the values of a convergent cantorian term [in W_n] is Pi_2. We can uniformize this Pi_2 relation by a Sigma_3 function. Applying Sigma_3 replacement, we get a single ordinal delta_n that bounds the values of all convergent terms in any instantiating model given by our current instantiating function. We can now define alpha_{n+1} it is the least lbfp which is greater than all of the following: (a) alpha_n; (b) gamma_{n+1}; (c) delta_n. We now argue much as in section 2.9 of Lemma 7, invoking Erdos-Rado, and getting an equivalence relation twiddle_eta [THAT DEPENDS ON eta] and a Sigma_2 function [analogous to the F_3 of the section cited] that gives for each eta, a model depending on eta, such that Y has ordertype at least eta, and that increasing tuples of the appropriate type instantiate the equivalence relation twiddle_eta. This only uses Sigma_2 uniformization. Here comes the crucial use of Sigma_3 replacement: We can find some fixed equivalence relation on the terms of rank at most (alpha_{n+1},n+1) that occurs as twiddle_eta for unboundedly many eta. [This uses Lemma 3 from letter 2 of phase B!]. Once we have this relation, we take it as b_{n+1}. Another application of Sigma_2-uniformization gets a Sigma_2 function defined on the infinite cardinals that well-instantiates b_{n+1}. This completes letter 3, and with it our discussion of phase B of the proof. From solovay@ccrwest.org Tue Mar 6 17:29:08 2001 Return-Path: Received: from diamond.boisestate.edu (IDENT:root@diamond.boisestate.edu [132.178.208.127]) by catseye.idbsu.edu (8.9.3/8.9.3) with ESMTP id RAA05716 for ; Tue, 6 Mar 2001 17:29:08 -0700 Received: from bluenote.ccrwest.org (smap@bluenote.ccrwest.org [192.203.205.129]) by diamond.boisestate.edu (8.9.3/8.9.3) with SMTP id RAA29301 for ; Tue, 6 Mar 2001 17:27:23 -0700 Received: by bluenote.ccrwest.org (4.1/CCRWEST-I1.20) id AA03236; Tue, 6 Mar 2001 16:27:21 PST Received: from ccrwest.ccrwest.org(192.203.205.65) by bluenote.ccrwest.org via smap (V1.3) id sma003234; Tue, 6 Mar 2001 16:27:05 -0800 Received: from grazia.ccrwest.org by ccrwest.ccrwest.org (4.1/CCRWEST-2.13) id AA12284; Tue, 6 Mar 2001 16:27:05 PST Received: from localhost by grazia.ccrwest.org (4.1/ccrwest-1.6) id AA10385; Tue, 6 Mar 2001 16:27:03 PST Date: Tue, 6 Mar 2001 16:27:03 -0800 (PST) From: "Robert M. Solovay" To: Randall Holmes Subject: Phase C: first letter Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO Randall, 1. Here starts the series of letters in which I present the proof that Con(ZC + Sigma_2 Replacement) entails Con(NFU*). The usual remarks about the metatheory apply. Officially, my metatheory is ZF-. But the proof would easily transcribe to Peano Arithmetic. I suspect that it would go through in PRA as well, but i definitely have not thought through the details that would establish this last claim. We are given a model N of ZC + Sigma_2 Replacement, N. We can and do assume that V=L holds in N. Moreover, we can and do assume that for no limit cardinal lambda of N is L_lambda a model of Sigma_2 replacement. It follows, as I discussed in the first letter of phase B, that there is a definable map from omega to OR, in N, whose range is cofinal in N. We can and do assume that this map is strictly increasing, and that its range consists of lbfp's. 2. There is much in common between the proof in phase C and the earlier proof in phase B, and I shall begin by reviewing these comment elements. Before doing that, I make the following remark: There is a clear lineage to this proof that runs as follows: Jensen's proof that for every ordinal alpha, there is an alpha model of NFU*. Phase A. Phase B. Phase C. Each phase has a lot in common with its immediate predecessor. But there is very little commonality [though there is some-- for example, the use of Erdos-Rado] between Jensen's original proof, and phase c. Thus had I chosen to present phase C without indicating its origins, it would look much more original than it actually is. 2. Here is a very high level outline of the proof. Again, we will be constructing a certain theory T [that plays the role of the "term model" M of phase B.]. The construction will take place in N, and proceed in omega stages. Roughly speaking, this has the effect that we are dealing with only a "set's worth" of problems at any stage. Because there is a definable cofinal map of omega into OR, we will succeed in "handling everything" by the end of our construction. We remark that the theory T will be a proper class of the model N. Indeed, even the approximations to T [call them T_n] will be proper classes of N. The T_n's will be Sigma_2, but we lost that in the limit, though T will be of course, definable in N. [I haven't bothered to compute what its complexity is. My guess is that it is either Sigma_3 or Sigma_4.] The theory T will have a canonical term-model M*. M* will be a model of a significant amount of set-theory: roughly, MacLane set-theory + KP + V=L + "There are arbitrarily large lbfps." The model M* will have an obvious automorphism and so yield, in the usual way, a model Q of NFU. We will take steps during our construction to ensure that the W [strongly cantorian initial segment of Z] of the model Q is canonically identified with the model N. This together with the fact that Q is a definable class of N will ensure, in the usual way, that Q is a model of NFU*. 3. Next let me emphasize what is different in the proof. Rather than working with a term language as I did in phases A and B I will be working in a certain infinitary language, rather like, but not exactly like the ones that figure in the Barwise compactness theorem. It turns out [and I only learned of this this past Sunday] that if V_kappa is a model ZC + Sigma_2 replacement and kappa has cofinality omega, then the infinitary logic based on V_kappa has strong compactness properties. [I will review this material in a subsequent letter.] One can apply these results "inside N". The "proofs" are sets of N, while the models constructed are proper classes of N. By using this machinery, we are able to carry out an analogue of the proof given in phase B for the weaker theory T_2. One thing that is peculiar about these results for V_kappa: The proofs of formal logic bear some resemblance to informal proofs. And the proofs involved in the usual Barwise compactness theorem bear a passing resemblance to formal proofs in first-order logic. But the "proofs" of the V_kappa framework don't look like proofs at all. This ends letter 1 of phase C. My next task is to present my version of the results about V_kappa. [Although it is all in Barwise's book "Admissible sets and structures", it took me a while to decode what he was saying. And when I did, I realized that I would present the material quite differently than he did.] From solovay@ccrwest.org Tue Mar 6 23:37:25 2001 Return-Path: Received: from diamond.boisestate.edu (IDENT:root@diamond.boisestate.edu [132.178.208.127]) by catseye.idbsu.edu (8.9.3/8.9.3) with ESMTP id XAA05999 for ; Tue, 6 Mar 2001 23:37:25 -0700 Received: from bluenote.ccrwest.org (smap@bluenote.ccrwest.org [192.203.205.129]) by diamond.boisestate.edu (8.9.3/8.9.3) with SMTP id XAA31764 for ; Tue, 6 Mar 2001 23:35:38 -0700 Received: by bluenote.ccrwest.org (4.1/CCRWEST-I1.20) id AA06775; Tue, 6 Mar 2001 22:35:36 PST Received: from ccrwest.ccrwest.org(192.203.205.65) by bluenote.ccrwest.org via smap (V1.3) id sma006773; Tue, 6 Mar 2001 22:35:17 -0800 Received: from opus.ccrwest.org by ccrwest.ccrwest.org (4.1/CCRWEST-2.13) id AA13417; Tue, 6 Mar 2001 22:35:17 PST Received: from localhost by opus.ccrwest.org (4.1/ccrwest-1.6) id AA15554; Tue, 6 Mar 2001 22:35:15 PST Date: Tue, 6 Mar 2001 22:35:15 -0800 (PST) From: "Robert M. Solovay" To: Randall Holmes Subject: Letter 2 of phase C Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO This letter is devoted to my exposition of the theory of compactness for uncountable structures V_kappa where kappa has cofinality omega. My source for all this is Barwise's book Admissible Sets and Structures". Particularly relevant is section 7 of Chapter VIII of Barwise. Especially Theorems 7.2 and 7.4. That said, I found Barwise's exposition of this needlessly long-winded and obscure. I hope to do a better job. 1. We will be working in ZFC. Our main focus is the following situation: (a) kappa is a cardinal of cofinality omega; (b) V_kappa is a model of ZC + Sigma_2-Replacement. As Barwise remarks, the arguments will go through in the following more general situation: (a) kappa is a cardinal of cofinality omega; (b) S is a subset of V_kappa; (c) P(x,y) is the relation : y is the power set of x; (d) define the notion of Delta_0 formula in the usual way except we allow atomic formulas of the form Sx and P(x,y) as well as those of the form x=y and x epsilon y. (e) The structure is admissible, Our special case corresponds to taking S to be the empty set. To get the general case, replace Sigma_2 by "Sigma_1 in the predicates P and S". 2. Our next task is to describe a certain infinitary language. The finiteness condition we impose is non-standard and is needed for our particular application. The general results I review in this letter would go through just as well if the finiteness condition were not imposed. Basic predicates of the language: Exactly as in set-theory, there are two: = and \epsilon. Functions: For each n in omega, there is a unary function h_n. These will play the same role as in the previous phases of the proof. Variables: It seems to me that I can get by with just countably many. But to be on the safe side, I will have a variable v_alpha for each ordinal alpha. Constants: These come in two flavors: 1) For each ordinal alpha [less than kappa-- we are working within V_kappa!] there will be a constant \bar{alpha}. The intended interpretation of \bar{alpha} is alpha. 2) For each ordinal alpha, there will be a constant xi_alpha. These play a somewhat analogous role to the xi_i's in the previous phases of the proof. And at a later point, I will need also xi_i's for negative integers i. But they won't appear in the current language definition. The definition of the language is inductive. In particular, there will be Henkin constants introduced by the following rule; If psi(x) is a formula having only the variable x free, then \iota x psi(x) is a constant of the language. Subsequently, I will need the notion of, for example, a xi_alpha appearing in some term or formula. One gives the obvious inductive definition. But in particular, if xi_alpha appears in the formula psi(x), it appears in the Henkin constant \iota x psi (x). The definition of term is evident as is that of atomic formula. [Though note well, that because of our use of Henkin constants, the notion of term and formula must really be given a simultaneous inductive definition.] The other clauses of the definition are: (a) if psi is a formula and x is a variable, then \forall x psi and \exists x psi are formulas. (b) if psi is a formula, so is \neg psi. (c) The final clause is a little more involved. We give first the usual version that is found in Barwise, and then the amendment that we will actually use. Imagine that \bigvee is the kind of "big V" used to denote a possibly infinite disjunction. And imagine that \bigand is a large upside down V of the sort used to denote a possibly infinite conjunction. The approximate version of clause (c) is: If S is a set of formulas, then <\bigvee, S> is a formula [and denotes the infinite disjunction of the formulas in S]; dually, If S is a set of formulas, then <\bigand, S> is a formula [and denotes the the infinite conjunction of the formulas in S]. CAUTION: When we say "set" here we really mean element of V_kappa. To get our precise definition, we need to impose three finiteness conditions on S. [The first is imposed by Barwise as well.] Finiteness conditions: 1) The set of alpha such that v_alpha occurs free in some formula of S is finite. 2) The set of alpha such that xi_alpha occurs in some formula of S is finite. 3) The set of n such that h_n appears in some formula of S is finite. Note well; We do not require that the set of alpha such that \bar{alpha} appears in some formula of S is finite. This completes our description of the syntax of our infinitary language L. I hope the semantics of L is evident. In particular, given a sentence of L [formula with no free variables] and an L-structure [a structure where all the constants and functions as well as the epsilon predicate are given interpretations], then the sentence has a truth-value relative to this structure which is defined in the evident way. Notice that the way we have set things up, the sentences, formulas, terms, etc, are elements of V_kappa. However, it will be important to allow structures which ar not elements of V_kappa. [We could get by considering just structures whose underlying sets are subsets of V_kappa.] It is fairly easy to see that the notions of formula, term, etc. are Delta_2(V_kappa). We Godel number the Sigma_2 subsets of V_kappa in an evident way. [I had to do essentially this when I Godel numbered Sigma_2 partial functions in letter 3 of phase B.] Let S be the set of sentences of L. [S is a "proper class' from the standpoint of V_kappa.] We get a Godel numbering of the Sigma_2 subsets of S as follows. The subset of S with Godel number e is obtained by intersecting S with the subset of V_kappa with Godel number e. [e here is some element of V_kappa.] Theorem: Sigma_2 completeness. The set of e such that the collection of sentences with Godel number e has a model [not necessarily in V_kappa!] is Pi_2. Theorem: Sigma_2 compactness. Let A be a Sigma_2 set of sentences of L. Then the following are equivalent: 1) A has a model. 2) For every a \subseteq A such that a \in V_kappa, a has a model. 3) For every a \subseteq A such that a \in V_kappa, a has a model which is an element of V_kappa. This completes this letter. These theorems are not at all evident. I will prove them in the next letter [or two]. From solovay@ccrwest.org Wed Mar 7 13:53:07 2001 Return-Path: Received: from diamond.boisestate.edu (IDENT:root@diamond.boisestate.edu [132.178.208.127]) by catseye.idbsu.edu (8.9.3/8.9.3) with ESMTP id NAA06691 for ; Wed, 7 Mar 2001 13:53:07 -0700 Received: from bluenote.ccrwest.org (smap@bluenote.ccrwest.org [192.203.205.129]) by diamond.boisestate.edu (8.9.3/8.9.3) with SMTP id NAA07544 for ; Wed, 7 Mar 2001 13:51:14 -0700 Received: by bluenote.ccrwest.org (4.1/CCRWEST-I1.20) id AA13532; Wed, 7 Mar 2001 12:51:08 PST Received: from ccrwest.ccrwest.org(192.203.205.65) by bluenote.ccrwest.org via smap (V1.3) id sma013519; Wed, 7 Mar 2001 12:50:08 -0800 Received: from makam.ccrwest.org by ccrwest.ccrwest.org (4.1/CCRWEST-2.13) id AA16984; Wed, 7 Mar 2001 12:50:08 PST Received: from localhost by makam.ccrwest.org (4.1/ccrwest-1.6) id AA12335; Wed, 7 Mar 2001 12:50:06 PST Date: Wed, 7 Mar 2001 12:50:06 -0800 (PST) From: "Robert M. Solovay" To: Randall Holmes Subject: C: letter 3 Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO Randall, Our current order of business is to prove the facts concerning infinitary logic that I asserted at the end of letter C2. Our approach will be as follows. 1) We shall see that a version of Konig's lemma holds in our current context [with "the usual proof"]. 2) It is well-known that the usual form of Konig's lemma is closely related to the Godel completeness theorem. A similar situation will obtain here and we shall prove the theorems on completeness and compactness asserted at the end of letter C2 from the current version of Konig's lemma. 2. Let's rehearse some well-known definitions. A tree T is a set |T| together with a partial-ordering, \leq_T such that the predecessors of any x in T are well-ordered by \leq_T. To avoid any ambiguity: \leq_T is reflexive. Let T be a tree. Each x in |T| has an ordinal attached, its level, which is the order-type of the set of y which are strictly less than x in the tree. I write level(alpha,T) for the set of points of T which appear at level alpha. 3. Next I have to explain when a tree T is "a Delta_2 tree". We require: 1) The underlying set of T is a Delta_2 subset of V_kappa. [So from the standpoint of V_kappa, T may be a proper class. 2) The partial-ordering of T is Delta_2. 3) The height of T is at most kappa. For each alpha < kappa, level(alpha, T) is an element of V_kappa. And the map which sens alpha to level(alpha, T) is Delta_2. 4. We can now formulate the version of Konig's lemma we will be proving. But first, I should probably recall our standing assumptions on kappa. 1) kappa is a limit cardinal of cofinality omega; 2) V_kappa is a model of ZC + Sigma_2-replacement. Here is the version of Konig's lemma we will be proving. Lemma 4. Let T be a Delta_2 tree. Suppose that for every alpha < kappa, level(alpha, T) is non-empty. Then T has a branch of order-type kappa. [A branch is a maximal linearly ordered subset of T.] Cf. Barwise op. cit. Chapter VIII, Theorem 7.2. Lemma 4 will follow immediately from the fact that kappa has cofinality omega and the following two claims; Claim 1. Let alpha < kappa. Then there is an x at level alpha in T such that for every beta > alpha, [with beta < kappa, of course] x has a descendent at level beta in T. Claim 2. Let x in T. We suppose that x has level alpha and that for every beta with alpha < beta < kappa, x has a descendent at level beta of the tree. Let beta_0 be given with alpha < beta_0 < kappa. Then x has a descendent y at level beta_0 with the following property: For every gamma with beta_0 < gamma < kappa, y has a descendent at level gamma in T. [When I say, e,g,. that x has a descendent y at level beta in T, I mean there is a y at level beta such that x \leq_T y.] The proofs of 1 and 2 are quite similar and I shall content myself with a proof of 2. But before beginning the proof, let me make a remark that I should have made earlier: ZC + Sigma_2-replacement proves Sigma_2-uniformization. [I. e., there is no need to require V=L.] The proof uses the fact that ZC + Sigma_2-Replacement proves: (a) For every alpha, V_alpha exists: (b) For every alpha, there is bfp greater than alpha. (c) [Levy absoluteness] If alpha is a bfp, V_alpha is absolute for Sigma_1 sentences. Let's begin the proof of claim 2. We are given x satisfying the hypotheses of claim 2. Towards a contradiction, assume that the conclusion of the lemma fails for some beta_0 > alpha. Let Y be the set of descendants of x at level beta_0. By assumption Y is non-empty. Introduce a Sigma_2 relation S(a,b) thus; a is in Y, b is an ordinal greater than beta_0 and a has no descendants at level b. Our assumptions imply that for every a in Y, there is a b such that S(a,b). Uniformize S to a Sigma_2 function F with domain Y. Let gamma be an ordinal greater than every ordinal in the range of F. By assumption, x has a descendant at level gamma, say z. Let y be the ancestor of z at level beta_0. Then y in Y, and gamma > F(y). But this contradicts the definition of F since y has the descendant z at level gamma. This completes our proof of the variant of Konig's Lemma and with it letter C3. From solovay@ccrwest.org Wed Mar 7 19:03:26 2001 Return-Path: Received: from diamond.boisestate.edu (IDENT:root@diamond.boisestate.edu [132.178.208.127]) by catseye.idbsu.edu (8.9.3/8.9.3) with ESMTP id TAA06836 for ; Wed, 7 Mar 2001 19:03:26 -0700 Received: from bluenote.ccrwest.org (smap@bluenote.ccrwest.org [192.203.205.129]) by diamond.boisestate.edu (8.9.3/8.9.3) with SMTP id TAA12951 for ; Wed, 7 Mar 2001 19:01:39 -0700 Received: by bluenote.ccrwest.org (4.1/CCRWEST-I1.20) id AA19111; Wed, 7 Mar 2001 18:01:32 PST Received: from ccrwest.ccrwest.org(192.203.205.65) by bluenote.ccrwest.org via smap (V1.3) id sma019105; Wed, 7 Mar 2001 18:01:02 -0800 Received: from opus.ccrwest.org by ccrwest.ccrwest.org (4.1/CCRWEST-2.13) id AA18852; Wed, 7 Mar 2001 18:01:03 PST Received: from localhost by opus.ccrwest.org (4.1/ccrwest-1.6) id AA17156; Wed, 7 Mar 2001 18:01:01 PST Date: Wed, 7 Mar 2001 18:01:01 -0800 (PST) From: "Robert M. Solovay" To: Randall Holmes Subject: Letter C4 Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO The goal of this letter is to use the variant of Konig's Lemma proved in letter C3 to establish the theorems asserted at the end of letter C2. 1. We introduced an infinitary language L in letter c2. Recall that S is the set of sentences of L. [From the standpoint of V_kappa, S is a proper class.] Let X be a Sigma_2 subset of S. We are going to associate to X a certain tree T(X). Roughly speaking, models of X will correspond to branches of T(X). T(X) will be a subtree of a certain universal tree, T*, which we build first. Work in V_kappa. We define a Delta_2 function, F, mapping kappa to kappa as follows: F(0) = 0. The ordinals F(1), F(2), ... enumerate in increasing order the lbfps less than kappa. [Implicit in this definition is the fact that there are kappa lbfps less than kappa.] We now describe the tree T*. The tree order will be inclusion. Level 0 of the tree will consist of one member, the empty set. Let alpha > 0. Let S_alpha consist of those sentences of L which lie in V(F(alpha)). Then the alpha^{th} level of T* consists of all maps of S_alpha into (0,1}. This completes the definition of T*. 2. We follow the usual conventions of thinking of 1 as "truth" and 0 as "falsehood". Very roughly, level alpha of T(X) will consist of those elements of level alpha of T* which "respect the logical connectives" and require those elements known to be in X at stage F(alpha) to get truth value 1. We turn to the precise requirements. So let alpha be given and let f be in level alpha of T*. We describe precisely the requirements that f must satisfy to be in level alpha of T(X). If alpha = 0, then the empty set will be in the bottom level of T(X). So assume from now on, that alpha > 0. Let the Sigma^2 definition of X have the form (exists x)A(x,s) where A is Pi_1. Requirement 1: Suppose that s \in V(F(alpha), s is a sentence, and for some x in V(F(alpha)) A(x,s). Then f(s) = 1. Requirement 2: [f behaves right on conjunctions]. Suppose that s is the conjunction of the set of sentences A. Suppose also that s \in S and s \in V(F(alpha)). Notice that this entails that each member of A is in S \cap V(F(alpha)), (a) If f(s) = 1, then for all a in A, f(a) = 1. (b) If f(s) = 0, then for some a in A, f(a) = 0. Requirement 3: [f behaves right on disjunctions.] Suppose that s is the disjunction of the set of sentences A. Suppose also that s \in S and s \in V(F(alpha)). Notice that this entails that each member of A is in S \cap V(F(alpha)), (a) If f(s) = 1, then for some a in A, f(a) = 1. (b) If f(s) = 0, then for all a in A, f(a) = 0. Requirement 4: [f behaves right on negations.] Let s be a sentence in S \cap V(F(alpha) and let t be the negation of s. Then f(t) = 1 - f(s). Requirement 5: [f behaves right on existential quantifiers.] (a) Let s be a sentence in the domain of f of the form "exists x psi(x)" Let t be a closed term of L lying in V(F(alpha)). Let s_1 be the sentence psi(t). Then if f(s_1) = 1, f(s) = 1. (b) Let s be a sentence in the domain of f of the form "exists x psi(x)". Let t_1 be the closed term \iota x psi(x) and let s_2 be the sentence "psi(t_1)". Then if f(s) = 1, then f(s_2) = 1. Requirement 6: [f behaves right on universal quantifiers]. Let s be a sentence in the domain of f of the form "forall x psi(x)". Let s_1 be the sentence \neg exists x \neg psi(x) Then f(s) = f(s_1). Notice that s_1 will be in the domain of f. Requirement 7: [f behaves right on "=".] (a). If s is is a sentence of the form "t = t" where t is a closed term, and s is in the domain of f, then f(s) = 1. (b) Suppose that t_1 and t_2 are closed terms in V(F(alpha)). If f("t_1 = t_2") = 1, then f("t_2 = t_1") = 1. (c) Suppose that t_1, t_2, and t_3 are closed terms in V(F(alpha)). Then if f("t_1 = t_2") = 1, and f("t_2 = t_3") = 1, then f("t_1 = t_3") = 1. (d) Suppose that t_1, t_2, t_3, t_4 are closed terms in V(F(alpha)). If f("t_1 = t_2") = 1, f("t_3 = t_4") = 1, and f("t_1 \in t_3") = 1, then f("t_2 \in t_4") = 1. (e) Let t_1, t_2 be closed terms in V(F(alpha)) and let n in omega. Then if f("t_1 = t_2") = 1, f("h_n(t_1) = h_n(t_2)") = 1. This completes our description of the tree T(X). Of course, we have just "done the obvious thing". It is evident that T(X) is a Delta_2 tree in the sense of letter C3. 3. Recall that a branch through a tree such as T(X) is a maximal linearly ordered subset that has order type kappa. So it has elements on every level. So if b is a branch through T(X), the union of b is a function mapping S to {0,1} that gives each s in X the value 1 and respects the various logical connectives. If M is a model of X, then M determines a function h:S --> {0,1} ["compute the truth value of the sentence s in M"]. The restriction of h to the various S_alpha's gives a branch through T(X). Conversely, if b is a branch through T(X), b determines a model of X as follows. First, we put an equivalence relation on the closed terms by saying t_1 == t_2 iff the union of b gives "t_1 = t_2" value 1. The underlying set of M is the set of equivalence classes. We define the other elements of the structure of M "according to b" in an evident way. Because of the requirements imposed on T(X), it is routine to check that a sentence is true in M iff the union of b gives it the value 1. [The fact that L is Henkenized is key here, of course.] The various claims at the end of letter C2 are now pretty evident. First, X has a model iff T(X) has a branch iff every level of T(X) is non-empty. But this last formulation is clearly Pi_2. As for compactness, obviously, if X has a model, then so does every bounded subset x of X which is in V_kappa. In fact, such a bounded subset will be in the domain of some member of the branch b that corresponds to the model [and which has rank > 0. But this member easily yields a model for x which is an element of V_kappa. Conversely, if every bounded subset of X has a model, level(alpha,T(X)) is non-empty for every alpha < kappa. Hence by Konig, T(X) has a branch, so X has a model. The various claims made at the end of letter C2 have been proved. This ends letter C4. From solovay@ccrwest.org Wed Mar 7 23:57:49 2001 Return-Path: Received: from diamond.boisestate.edu (IDENT:root@diamond.boisestate.edu [132.178.208.127]) by catseye.idbsu.edu (8.9.3/8.9.3) with ESMTP id XAA06923 for ; Wed, 7 Mar 2001 23:57:49 -0700 Received: from bluenote.ccrwest.org (smap@bluenote.ccrwest.org [192.203.205.129]) by diamond.boisestate.edu (8.9.3/8.9.3) with SMTP id XAA14097 for ; Wed, 7 Mar 2001 23:55:55 -0700 Received: by bluenote.ccrwest.org (4.1/CCRWEST-I1.20) id AA21038; Wed, 7 Mar 2001 22:55:39 PST Received: from ccrwest.ccrwest.org(192.203.205.65) by bluenote.ccrwest.org via smap (V1.3) id sma021036; Wed, 7 Mar 2001 22:54:40 -0800 Received: from opus.ccrwest.org by ccrwest.ccrwest.org (4.1/CCRWEST-2.13) id AA19493; Wed, 7 Mar 2001 22:54:41 PST Received: from localhost by opus.ccrwest.org (4.1/ccrwest-1.6) id AA17404; Wed, 7 Mar 2001 22:54:39 PST Date: Wed, 7 Mar 2001 22:54:39 -0800 (PST) From: "Robert M. Solovay" To: Randall Holmes Subject: Letter C5 Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO The main purpose of this [hopefully brief] letter is to discuss what the compactness and completeness results proved in the last two letters look like when formulated "internal to our model N". But first I want to correct an unimportant misstatement that I made. I said in letter C2, as I recall, that the "proofs" for the compactness results formulated for V_kappa in C_2 look nothing like those for the usual Barwise compactness. But in fact they can be made to look quite similar. One basically has one new type of axiom saying that if S is a set of sentences, then any truth assignment to S lies in the ground model. [This can be expressed as a big infinitary disjunction.] In our current context, this is a little awkward to carry out since the new axioms don't meet our finiteness restraints. This difficulty can be circumvented, but since there is no particular need to do so I shall say no more about this. 2. So on to our main topic. We have a model N of T_2 + "For no limit cardinal lambda is L_lambda a model of T_2". As I have remarked previously, it follows that there is a definable map from omega to OR, in N, whose range is cofinal in N. We let the map send i to gamma_i; we shall assume that the gamma_i's are lbfps and that the map i --> gamma_i is strictly increasing. There is no difficulty carrying out the definition of our language L internal to N. Where we previously had, for example, a constant \bar{alpha} for each alpha < kappa, we now have such a constant for each ordinal alpha. The collections of terms, sentences, etc. are proper classes which are Delta_2. If X is a Sigma_2 set of sentences of L, we can form the tree T(X) much as before. It will once again be a Delta_2 tree. Konig's lemma now takes the following form: If every level of T(X) is non-empty then there is a definable branch through T. The old proof goes through, mutatis mutandis, using in a crucial way that there is a definable map that shows that OR has cofinality omega. Such a branch through T(X) yields a canonical model M of X. I will make one small change in the definition of M. Each equivalence class of closed terms has a least member with respect to the canonical well-ordering of N. [Recall that N is a model of V=L.] I will take these least members rather than the equivalence classes themselves as the elements of the underlying class of M. [In general, M will be given as a proper class of the model N.] If one is being truly pedantic, there are proper classes which give; (a) the underlying class of M; (b) The map from the class of closed terms of L onto M; (c) The class of sentences of L which are true. This gives us a quite satisfactory grip on M. The completeness theorem holds in the form that the class of Godel numbers [as Sigma_2 subsets of the class of sentences S] of X's that have models [in the sense that T(X) has a branch] is Pi^2. All that really matters is that this is a definable class in N. The precise hierarchy calculation does not matter. The compactness theorem goes through as before: X "has a model" [in the sense that T(X) has a branch] iff every subset x of X has a set-model. There is one small point in the proof of this that I slurred over in the last letter. Let x be a subset of X. Then using Sigma_2-replacement one sees that there is some stage alpha [which is an lbfp] at which all the Sigma_2 facts that show each member of x actually is in X have witnesses in V(alpha). This is needed to get a model of x from an appropriate node of the tree T(X). This completes my discussion of the problem of internalizing the results of Letters C3 and C4. End of letter C5. From solovay@ccrwest.org Thu Mar 8 13:41:04 2001 Return-Path: Received: from diamond.boisestate.edu (IDENT:root@diamond.boisestate.edu [132.178.208.127]) by catseye.idbsu.edu (8.9.3/8.9.3) with ESMTP id NAA07452 for ; Thu, 8 Mar 2001 13:41:03 -0700 Received: from bluenote.ccrwest.org (smap@bluenote.ccrwest.org [192.203.205.129]) by diamond.boisestate.edu (8.9.3/8.9.3) with SMTP id NAA21660 for ; Thu, 8 Mar 2001 13:39:10 -0700 Received: by bluenote.ccrwest.org (4.1/CCRWEST-I1.20) id AA27485; Thu, 8 Mar 2001 12:39:00 PST Received: from ccrwest.ccrwest.org(192.203.205.65) by bluenote.ccrwest.org via smap (V1.3) id sma027483; Thu, 8 Mar 2001 12:38:43 -0800 Received: from grazia.ccrwest.org by ccrwest.ccrwest.org (4.1/CCRWEST-2.13) id AA22288; Thu, 8 Mar 2001 12:38:44 PST Received: from localhost by grazia.ccrwest.org (4.1/ccrwest-1.6) id AA12575; Thu, 8 Mar 2001 12:38:42 PST Date: Thu, 8 Mar 2001 12:38:42 -0800 (PST) From: "Robert M. Solovay" To: Randall Holmes Subject: Letter C6 Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO This letter starts the main thrust of the argument. Let me outline how it will go. We have our model N of T_2 whose properties I have just recalled in letter C5. Our construction will take place "inside N". The main work is a length omega construction. One of the things that we will be building, step by step, is a series of Sigma_2 theories in our language L, T_0, T_1, ... This series of theories is increasing. Its union T* need not be Sigma_2, but it will certainly be a class of N. It will be evident from our construction that T* is complete. Moreover, it will be true for the limit theory that the xi_alpha's are indiscernibles. We will build the model M* according to the blueprint given by T* but using as generating indiscernibles the xi_n for n in Z [Here Z is the set of all integers both positive and negative as computed in N.] M* has an evident automorphism which sends xi_i to xi_{i+1} for any i. M* will be a model of a moderately strong set-theory and the xi_n's will be lbfps in that model, so we will get a model Q of NFU in the usual way. It will be evident from our construction that the strongly cantorian ordinals of Q are canonically identified with the ordinals of N. In the usual way, this plus the fact that Q is given by a class of N will ensure that the crucial axioms of counting and strongly Cantorian full selection hold in Q so that Q is indeed a model of NFU*. 2. The next order of business is to describe the theory T_0 and prove it has a model. [I always mean by this that the relevant tree T(T_0) has a branch.] Most of the axioms of T_0 are routine, but there is at least one tricky one that I will call attention to when we get to it. Axiom group 1: Axioms of set-theory We can find finitely many axioms in the usual language of set-theory [first-order logic is a subset of L!] that express: (a) Maclane set-theory; (b) KP; (c) V=L; (d) There are arbitrarily large lbfps. (e) For no limit cardinal lambda is L_lambda a model of Sigma_2-Replacement. [I think I don't actually need axiom 1(e), but it doesn't hurt.] Axiom group 2: Axioms on the \bar{alpha}'s. For each alpha in OR, there is an axiom that says: For all x [x is in \bar{alpha} iff \bigvee {x = \bar{beta} | beta < alpha} [This is the usual way one pins down the meaning of \bar{alpha} in infinitary logic.] Axiom group 3: Axioms on the xi_alpha's. For each ordinal alpha, there is an axiom: xi_alpha is an lbfp. For each pair of ordinals, alpha, beta with alpha < beta, there is an axiom: xi_alpha \in xi_beta Axiom group 4: Axioms on the h_n's. (a) If x is not an ordinal, h_n(x) = 0. (b) If x is an ordinal, h_n(x) <= x. There is one instance of each axiom for each n in omega. Axiom group 5: The "no new sets" axioms. [By the way, I believe these axioms are unnecessary. I am including them "to be safe".] Let S be a set of sentences. We suppose that there is a finite subset of omega, A, and a finite subset of OR, B such that: (a) if h_n appears in some s in S, then n \in A. (b) If xi_alpha appears in some s in S, then alpha \in B. We have to construct some infinitary formulae. Lets start with the following: If theta is a formula of L, then by theta^0 we shall mean \neg theta . theta^1 will just mean theta. Now let f:S --> {0,1}. We introduce the formula A(S,f) which roughly says that f correctly describes the truth values of sentences in S: \bigand {theta^{f(theta)} | theta \in S} We can now formulate the instance of the "no new sets" axiom corresponding to S, B(S): \bigvee {A(S,f) | f: S --> {0,1}}. So there is one such axiom in T_0 for each S that meets our finiteness constraints. Axiom group 6: The least ordinal principle The purpose of these axioms is to enforce, to the extent that we can, that the models of T_0 are well-founded. [We definitely don't fully succeed. One can prove that the ordinals of any model of T_0 are not well-ordered.] Let X be a set of closed terms of L. There will be one instance of the axiom for each such X that meets the following finiteness constraint: There is a finite subset of omega, A, and a finite subset of OR, B such that: (a) if h_n appears in some x in X, then n \in A. (b) If xi_alpha appears in some x in X, then alpha \in B. Our axiom will have the form "If H(X) then C(X)". We describe these two components in turn: H(X) will express that all the members of x are ordinals: \bigand {x is an ordinal | x \in X} C(X) will express that some member of X is least: To start, let [for x in X], D(x,X) express that x is least in X: \bigand { x <= y | y in X} Then C(X) is just the obvious infinite disjunction: \bigvee {D(x,X) | x \in X} This completes our description of the theory T_0. It is obviously Sigma^2. In fact, it is obviously Delta_2. I say that T_0 is consistent [in the usual meaning we have been giving to such phrases that the corresponding tree T(T_0) has a branch through it]. For this, its enough, by compactness, to check that every set of axioms of T_0 has a model. But this is easy. For any set, a, of axioms of T_0, it is easy to whip up a model whose underlying set is L_lambda, where lambda = F(gamma^+). Here F is the function from letter C4 that enumerates the lbfps; gamma^+ is the least cardinal greater than gamma; and gamma \in OR is chosen sufficiently large compared to a. This is a good place to pause and I will end letter C6 here. From solovay@ccrwest.org Thu Mar 8 15:39:59 2001 Return-Path: Received: from diamond.boisestate.edu (IDENT:root@diamond.boisestate.edu [132.178.208.127]) by catseye.idbsu.edu (8.9.3/8.9.3) with ESMTP id PAA07891 for ; Thu, 8 Mar 2001 15:39:59 -0700 Received: from bluenote.ccrwest.org (smap@bluenote.ccrwest.org [192.203.205.129]) by diamond.boisestate.edu (8.9.3/8.9.3) with SMTP id PAA25883 for ; Thu, 8 Mar 2001 15:38:11 -0700 Received: by bluenote.ccrwest.org (4.1/CCRWEST-I1.20) id AA28979; Thu, 8 Mar 2001 14:38:04 PST Received: from ccrwest.ccrwest.org(192.203.205.65) by bluenote.ccrwest.org via smap (V1.3) id sma028972; Thu, 8 Mar 2001 14:37:44 -0800 Received: from opus.ccrwest.org by ccrwest.ccrwest.org (4.1/CCRWEST-2.13) id AA22812; Thu, 8 Mar 2001 14:37:44 PST Received: from localhost by opus.ccrwest.org (4.1/ccrwest-1.6) id AA18622; Thu, 8 Mar 2001 14:37:43 PST Date: Thu, 8 Mar 2001 14:37:42 -0800 (PST) From: "Robert M. Solovay" To: Randall Holmes Subject: Letter C7 Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO Randall, This letter handles a number of minor points that have come up. I will return to the main thrust of the argument for Phase C in the next letter. 1. In letter C3, I wrote: Uniformize S to a Sigma_2 function F with domain Y. But in that context, V_kappa does not have a canonical well-ordering so not all relations can be uniformized. We are, in fact, ok since the codomain of the relation in question consists of ordinals. 2. In stating my metatheory [ZF-] I remarked that it was not clear to me the relative consistency proofs I was giving could be carried out in primitive recursive arithmetic. While I still am reluctant to make an official claim, I've thought about things some more, and it looks routine to carry out the proofs in even the much weaker system IDelta_0 + "Exponentiation is total". [This system is too weak to carry out the equiconsistency of ZF and GB.] 3. I've been somewhat sloppy about the formal details of our length omega constructions. [One occurred in phase B, and one will be about to be presented in phase C.] What's going on is the following: There is some inductive condition I(x,n). There is some next-step condition S(x,y,n). We have the following holding in our model N: 1) There is an x such that I(x,0). 2) For every n in omega, for every x: If I(x,n) there is a y such that S(x,y,n) and I(y, n+1). In that case there is a class of N which is a function f with domain omega such that: 1) For all n, I(f(n),n); 2) For all n, S(f(n), f(n+1), n). Roughly, one takes f(0) to be the L-least x such that I(x,0). One takes f(n+1) to be the L-least y such that S(f(n),y,n) and I(y, n+1). I haven't been too explicit about spelling out I and S precisely, but they are there in the background and would need to be fully spelled out in a completely formal proof. This ends my series of brief remarks and with it letter C7. From solovay@yahoo.com Tue Mar 13 11:31:14 2001 Return-Path: Received: from diamond.boisestate.edu (IDENT:root@diamond.boisestate.edu [132.178.208.127]) by catseye.idbsu.edu (8.9.3/8.9.3) with ESMTP id LAA12770 for ; Tue, 13 Mar 2001 11:31:13 -0700 Received: from web1602.mail.yahoo.com (web1602.mail.yahoo.com [128.11.23.202]) by diamond.boisestate.edu (8.9.3/8.9.3) with SMTP id LAA18770 for ; Tue, 13 Mar 2001 11:29:18 -0700 Received: (qmail 14938 invoked by uid 60001); 13 Mar 2001 18:29:13 -0000 Message-ID: <20010313182913.14937.qmail@web1602.mail.yahoo.com> Received: from [152.163.213.64] by web1602.mail.yahoo.com; Tue, 13 Mar 2001 10:29:13 PST Date: Tue, 13 Mar 2001 10:29:13 -0800 (PST) From: Robert Solovay Subject: Re: letters on NFU* To: "M. Randall Holmes" In-Reply-To: <200103131717.KAA12656@catseye.idbsu.edu> MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Status: RO Randall, I'm preparing revised versions of the letters in group C. In addition to sending you the revised versions, I'll mail you context diffs so you can see where and what changes I've made. --Bob --- "M. Randall Holmes" wrote: > > Dear Bob, > > This is just to tell you that I have made a first > pass at reading > the recent letters on proving the consistency of > NFU* I have no questions > as yet, but this is not the same as saying that I > follow you yet! > I shall go and check out the local copy of > Admissible Sets and > Structures and get to work... > > --Randall __________________________________________________ Do You Yahoo!? Yahoo! Auctions - Buy the things you want at great prices. http://auctions.yahoo.com/ From solovay@ccrwest.org Tue Mar 13 15:21:13 2001 Return-Path: Received: from diamond.boisestate.edu (IDENT:root@diamond.boisestate.edu [132.178.208.127]) by catseye.idbsu.edu (8.9.3/8.9.3) with ESMTP id PAA13037 for ; Tue, 13 Mar 2001 15:21:13 -0700 Received: from bluenote.ccrwest.org (smap@bluenote.ccrwest.org [192.203.205.129]) by diamond.boisestate.edu (8.9.3/8.9.3) with SMTP id PAA23629 for ; Tue, 13 Mar 2001 15:19:16 -0700 Received: by bluenote.ccrwest.org (4.1/CCRWEST-I1.20) id AA23558; Tue, 13 Mar 2001 14:19:07 PST Received: from ccrwest.ccrwest.org(192.203.205.65) by bluenote.ccrwest.org via smap (V1.3) id sma023555; Tue, 13 Mar 2001 14:18:22 -0800 Received: from grazia.ccrwest.org by ccrwest.ccrwest.org (4.1/CCRWEST-2.13) id AA11931; Tue, 13 Mar 2001 14:18:22 PST Received: from localhost by grazia.ccrwest.org (4.1/ccrwest-1.6) id AA17531; Tue, 13 Mar 2001 14:18:20 PST Date: Tue, 13 Mar 2001 14:18:20 -0800 (PST) From: "Robert M. Solovay" To: Randall Holmes Subject: Changes in C letters Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO Randall, What follows is a brief description of the changes I'm making in section C. For the most part they are cosmetic. 1) Instead of one-infinitary language L, there will be a language L_alpha for each alpha <= omega. L_omega is [subject to the minor changes described below] the same as the old L. L_n differs only in that h_i is a function symbol of L_n iff i < n. The theory T_n is in the language L_n. 2. In the official language L_alpha, there are only infinitary disjunctions and \exists. [\forall and infinitary conjunctions are introduced as derived notions in the usual way.] 3. I spell out quite specifically what set corresponds to each term or formula. This is done in a totally routine way. I do this now since I want to talk about the terms or formula which are members of a particular V(alpha) at some point. Any reasonable implementation would work for what I need to do, but I thought I would provide a specific one. 4. There are two axioms of T_0 that I commented I didn't really need. I've now deleted them. 5. I formally introduce the notion of xi-support and h-support of a term or formula. The xi-support, for example, is the set of alpha such that xi_alpha appears in the term or formula in question. As I said, pretty minor changes. --Bob From solovay@ccrwest.org Wed Mar 14 08:05:40 2001 Return-Path: Received: from diamond.boisestate.edu (IDENT:root@diamond.boisestate.edu [132.178.208.127]) by catseye.idbsu.edu (8.9.3/8.9.3) with ESMTP id IAA00633 for ; Wed, 14 Mar 2001 08:05:40 -0700 Received: from bluenote.ccrwest.org (smap@bluenote.ccrwest.org [192.203.205.129]) by diamond.boisestate.edu (8.9.3/8.9.3) with SMTP id XAA27803 for ; Tue, 13 Mar 2001 23:09:21 -0700 Received: by bluenote.ccrwest.org (4.1/CCRWEST-I1.20) id AA27808; Tue, 13 Mar 2001 22:09:20 PST Received: from ccrwest.ccrwest.org(192.203.205.65) by bluenote.ccrwest.org via smap (V1.3) id sma027806; Tue, 13 Mar 2001 22:08:48 -0800 Received: from makam.ccrwest.org by ccrwest.ccrwest.org (4.1/CCRWEST-2.13) id AA13329; Tue, 13 Mar 2001 22:08:48 PST Received: from localhost by makam.ccrwest.org (4.1/ccrwest-1.6) id AA20359; Tue, 13 Mar 2001 22:08:46 PST Date: Tue, 13 Mar 2001 22:08:46 -0800 (PST) From: "Robert M. Solovay" To: Randall Holmes Subject: Letter C8 Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO So lets start the inductive construction: For each stage n in omega, we are going to define, by induction on n: (a) A consistent Sigma_2-theory T_n in the language L_n. We will have T_n extends T_{n-1} if n > 0. Really, we are inductively choosing the Godel number of T_n, which is a set. T_n itself, of course, is a proper class. (b) An ordinal alpha_n. alpha_0 will be 0. If n > 0, then alpha_n will be a lbfp which is greater than alpha_{n-1} and gamma_n. [The gamma's are our standard sequence for cofinalizing OR.] (c) A function f_n whose domain is the set of all sentences s of the language L_n such that: (1) s \in V(alpha_n); (2) If xi_alpha appears in s, then alpha < n. Thus f_0 will be the empty function. If f_n(s) = 1, s will be an axiom of T_n; if f_n(s) = 0, \neg s will be an axiom of T_n We will require that T_n contains the following indiscernibility axioms: Let s be in the domain of f_n. Let be an increasing n-tuple of ordinals. Let s' be obtained from s by replacing xi_i by xi_{alpha_i} uniformly throughout s. [This has to be explicated by a straighforward inductive definition.] Then "s iff s' " is an axiom of T_n. In order to keep things rolling we need to inductively maintain the following property of the T_n's: Let lambda_1 and lambda_2 be lbfps with alpha_n < lambda_1 < lambda_2. Let t_i be the intersection of T_n with V(lambda_i) for i = 1,2. Let M be a set model of t_2. Let f: lambda_1 --> lambda_2 be order preserving. f determines a map f* of the terms and formulas of L_n \cap V(lambda_1) into the terms and formulas of L_n \cap V(lambda_2) as follows: The map is defined inductively. It leaves all the language elements alone except the xi_alphas. It replaces xi_alpha by xi_{f(alpha)}. We convert the model M into a premodel M^\star for the language L_n \cap V(lambda_1) as follows: The meanings of \in and = do not change. Nor do the meanings of the \bar{alpha}'s or the h_i's. For a general closed term t, the meaning of t in M^\star is the meaning of f*(t) in M. Our key assumption [the thinning assumption] is that M^\star is a model of t_1. We will also have to maintain inductively that T_n "is consistent". [That is, T(T_n) has a branch of order type OR. By Konig this is equivalent to all the levels of T(T_n) being non-empty which can be expressed "inside N".] Here is a rough outline of the passage from T_n to T_{n+1}. 1) We will examine the "Cantorian terms of rank at most (alpha_{n}, n)". [The phrase in quotes will be defined presently.] We will separate these into convergent and potentially divergent terms. We will take action to make the potentially divergent terms divergent. The result of our actions will be a consistent theory, T_{n,1} in the language L_n. 2) We will add some new axioms concerning h_n. This will require the transition from the language L_n to L_{n+1}. The axioms will ensure that the divergent Cantorian terms will not be strongly Cantorian in the final model. The resulting theory is T_{n,2}. 3) At this stage, we will be able to choose alpha_{n+1}. It will be evident that all the convergent Cantorian terms will have denotations below alpha_{n+1}. 4) We next add indiscernibility sentences for n+1-tuples of xi_alpha's. The resulting theory is T_{n,3}. In checking that T_{n,3} is consistent, use will be made of our "thinning assumption" and Erdos-Rado. 5) Finally, we will decide the truth or falsity of any sentence of rank at most (alpha_{n+1}, n+1). Adding the sentences which reflect our decisions will give us T_{n+1}. Of course, we must be sure that T_{n+1} is consistent and satisfies our "thinning assumption". This is a good place to end letter C8. In the next letter[s] will will carry out our inductive construction in detail. *** holmes Tue Mar 13 21:16:20 2001 --- holmes.new Tue Mar 13 17:54:20 2001 *************** *** 22,27 **** --- 22,29 ---- well, but i definitely have not thought through the details that would establish this last claim. + [But see my comments re this in letter C7.] + We are given a model N of ZC + Sigma_2 Replacement, N. We can and do assume that V=L holds in N. Moreover, we can and do assume that for no limit cardinal lambda of N is L_lambda a *************** *** 29,35 **** It follows, as I discussed in the first letter of phase B, that there is a definable map from omega to OR, in N, whose ! range is cofinal in N. We can and do assume that this map is strictly increasing, and that its range consists of lbfp's. 2. There is much in common between the proof in phase C and --- 31,37 ---- It follows, as I discussed in the first letter of phase B, that there is a definable map from omega to OR, in N, whose ! range is cofinal in OR. We can and do assume that this map is strictly increasing, and that its range consists of lbfp's. 2. There is much in common between the proof in phase C and *************** *** 51,57 **** Each phase has a lot in common with its immediate predecessor. But there is very little commonality [though there is some-- for example, the use of Erdos-Rado] between ! Jensen's original proof, and phase c. Thus had I chosen to present phase C without indicating its origins, it would look much more original than it actually is. --- 53,59 ---- Each phase has a lot in common with its immediate predecessor. But there is very little commonality [though there is some-- for example, the use of Erdos-Rado] between ! Jensen's original proof, and phase C. Thus had I chosen to present phase C without indicating its origins, it would look much more original than it actually is. *************** *** 67,74 **** We remark that the theory T will be a proper class of the model N. Indeed, even the approximations to T [call them T_n] will be proper classes of N. The T_n's will be Sigma_2, but we ! lost that in the limit, though T will be of course, definable ! in N. [I haven't bothered to compute what its complexity is. My guess is that it is either Sigma_3 or Sigma_4.] --- 69,76 ---- We remark that the theory T will be a proper class of the model N. Indeed, even the approximations to T [call them T_n] will be proper classes of N. The T_n's will be Sigma_2, but we ! will lose that in the limit, though T will be of course, ! definable in N. [I haven't bothered to compute what its complexity is. My guess is that it is either Sigma_3 or Sigma_4.] *************** *** 89,95 **** Rather than working with a term language as I did in phases A and B I will be working in a certain infinitary language, ! rather like, but not exactly like the ones that figure in the Barwise compactness theorem. It turns out [and I only learned of this this past Sunday] --- 91,97 ---- Rather than working with a term language as I did in phases A and B I will be working in a certain infinitary language, ! rather like, but not exactly like, the ones that figure in the Barwise compactness theorem. It turns out [and I only learned of this this past Sunday] *************** *** 110,115 **** --- 112,123 ---- proofs in first-order logic. But the "proofs" of the V_kappa framework don't look like proofs at all. + Since writing the previous paragraph I have figured out how to + modify the usual Barise logic so as to apply to the V_kappa's + that we are considering in this series of lemmas. But the + point has no relevance to our current goals so I won't present + it. + This ends letter 1 of phase C. My next task is to present my version of the results about V_kappa. [Although it is all in Barwise's book "Admissible sets and structures", it took me a *************** *** 171,195 **** set. To get the general case, replace Sigma_2 by "Sigma_1 in the predicates P and S". ! 2. Our next task is to describe a certain infinitary ! language. The finiteness condition we impose is non-standard and is needed for our particular application. The general results I review in this letter would go through just as well if the finiteness condition were not imposed. ! Basic predicates of the language: Exactly as in set-theory, there are two: = and \epsilon. Functions: ! For each n in omega, there is a unary function h_n. These will play the same role as in the previous phases of the proof. ! Variables: It seems to me that I can get by with just ! countably many. But to be on the safe side, I will have a ! variable v_alpha for each ordinal alpha. Constants: These come in two flavors: --- 179,207 ---- set. To get the general case, replace Sigma_2 by "Sigma_1 in the predicates P and S". ! ! 2. Our next task is to describe a certain family of infinitary ! languages. The finiteness condition we impose is non-standard and is needed for our particular application. The general results I review in this letter would go through just as well if the finiteness condition were not imposed. ! There will be one language L_alpha for each alpha \leq omega. + Basic predicates of the language L_alpha: + Exactly as in set-theory, there are two: = and \epsilon. Functions: ! For each n < alpha, there is a unary function h_n. These will play the same role as in the previous phases of the proof. ! Variables: + There will be a variable v_alpha for each ordinal alpha. + + Constants: These come in two flavors: *************** *** 218,223 **** --- 230,247 ---- appears in the formula psi(x), it appears in the Henkin constant \iota x psi (x). + Some comments: Barwise works with Skolem functions rather than + Henkin constants. Normally that would be the right thing to + do. But for our purposes, the use of Henkin constants proves + more convenient. + + We define the xi-support of a term or formula to be the set of + alpha such that xi_alpha appears in the term or formula. + + The notion of the h-support of a term or formula is + analogous. It is the set of n such that h_n appears in the + term or formula. + The definition of term is evident as is that of atomic formula. [Though note well, that because of our use of Henkin constants, the notion of term and formula must really be given *************** *** 225,232 **** The other clauses of the definition are: ! (a) if psi is a formula and x is a variable, then \forall x ! psi and \exists x psi are formulas. (b) if psi is a formula, so is \neg psi. --- 249,256 ---- The other clauses of the definition are: ! (a) if psi is a formula and x is a variable, then \exists x ! psi is a formula. (b) if psi is a formula, so is \neg psi. *************** *** 244,252 **** formulas, then <\bigvee, S> is a formula [and denotes the infinite disjunction of the formulas in S]; - dually, If S is a set of formulas, then <\bigand, S> is a - formula [and denotes the the infinite conjunction of the - formulas in S]. CAUTION: When we say "set" here we really mean element of V_kappa. --- 268,273 ---- *************** *** 272,277 **** --- 293,305 ---- This completes our description of the syntax of our infinitary language L. + We have chosen not to have \forall and infinite conjunctions + to be an official part of our infinitary languages. However, + they can be introduced by abbreviations [inspired by de + Morgan's laws] in the usual way. We will feel free to use them + when informally describing formulas. + + I hope the semantics of L is evident. In particular, given a sentence of L [formula with no free variables] and an L-structure [a structure where all the constants and functions *************** *** 285,290 **** --- 313,351 ---- V_kappa. [We could get by considering just structures whose underlying sets are subsets of V_kappa.] + I am next going to explicitly spell out what set corresponds + to each term or formula. The definition is inductive; we let + <> be the set corresponding to the term or formula s. + + The definition that follows is totally routine. There is + nothing tricky going on. + + <> = <0,alpha>; + + <> = <1,alpha>; + + <<\bar{alpha}>> = <2, alpha>; + + << \iota x psi >> = <3, <>, <> >; + + << h_n (t) >> = <4, n, <> >; + + << \exists x psi >> = <5, <>, <> >; + + << \neg psi >> = <6, <> > ; + + Now let S be a set of formulas. Let S* be {<> | s \in S}. + + << \bigvee S >> = <7, S* > ; + + << s \in t >> = <8, <>, <> > + + << s = t >> = <9, <>, <> > + + I don't think I will ever use explicitly the definition just + given. But it is already being implicitly used in the + immediately following paragraph. + It is fairly easy to see that the notions of formula, term, etc. are Delta_2(V_kappa). *************** *** 320,325 **** --- 381,388 ---- + + From solovay@ccrwest.org Wed Mar 7 12:50:07 2001 Date: Wed, 7 Mar 2001 12:50:06 -0800 (PST) From: "Robert M. Solovay" *************** *** 371,377 **** 3) The height of T is at most kappa. For each alpha < kappa, level(alpha, T) is an element of V_kappa. And the map which ! sens alpha to level(alpha, T) is Delta_2. 4. We can now formulate the version of Konig's lemma we will be proving. But first, I should probably recall our standing --- 434,440 ---- 3) The height of T is at most kappa. For each alpha < kappa, level(alpha, T) is an element of V_kappa. And the map which ! sends alpha to level(alpha, T) is Delta_2. 4. We can now formulate the version of Konig's lemma we will be proving. But first, I should probably recall our standing *************** *** 416,424 **** myself with a proof of 2. But before beginning the proof, let me make a remark that I should have made earlier: ! ZC + Sigma_2-replacement proves Sigma_2-uniformization. ! [I. e., there is no need to require V=L.] The proof uses the fact that ZC + Sigma_2-Replacement proves: (a) For every alpha, V_alpha exists: --- 479,497 ---- myself with a proof of 2. But before beginning the proof, let me make a remark that I should have made earlier: ! ZC + Sigma_2-replacement proves the following special case of ! Sigma_2-uniformization. [I. e., there is no need to require V=L.] + Let R(x,y) be a Sigma_2-relation. We suppose that whenever + R(a,b) obtains then b is an ordinal. Then there is a + Sigma_2-relation S such that: + + (1) If S(a,b), then R(a,b). + + (2) If S(a,b) and S(a,b'), then b = b'. + + (3) If R(a,b), then for some b', S(a,b'). + The proof uses the fact that ZC + Sigma_2-Replacement proves: (a) For every alpha, V_alpha exists: *************** *** 477,485 **** 1. ! We introduced an infinitary language L in letter c2. Recall ! that S is the set of sentences of L. [From the standpoint of ! V_kappa, S is a proper class.] Let X be a Sigma_2 subset of S. We are going to associate to X a certain tree T(X). Roughly speaking, models of X will correspond to branches of T(X). --- 550,558 ---- 1. ! Let L be one of the languages L_alpha introduced in letter ! C2. Let S be the set of sentences of L. [From the standpoint ! of V_kappa, S is a proper class.] Let X be a Sigma_2 subset of S. We are going to associate to X a certain tree T(X). Roughly speaking, models of X will correspond to branches of T(X). *************** *** 530,548 **** Then f(s) = 1. ! Requirement 2: [f behaves right on conjunctions]. - Suppose that s is the conjunction of the set of sentences A. - Suppose also that s \in S and s \in V(F(alpha)). Notice - that this entails that each member of A is in S \cap - V(F(alpha)), - - (a) If f(s) = 1, then for all a in A, f(a) = 1. - - (b) If f(s) = 0, then for some a in A, f(a) = 0. - - Requirement 3: [f behaves right on disjunctions.] - Suppose that s is the disjunction of the set of sentences A. Suppose also that s \in S and s \in V(F(alpha)). Notice that this entails that each member of A is in S \cap --- 603,610 ---- Then f(s) = 1. ! Requirement 2: [f behaves right on disjunctions.] Suppose that s is the disjunction of the set of sentences A. Suppose also that s \in S and s \in V(F(alpha)). Notice that this entails that each member of A is in S \cap *************** *** 552,563 **** (b) If f(s) = 0, then for all a in A, f(a) = 0. ! Requirement 4: [f behaves right on negations.] Let s be a sentence in S \cap V(F(alpha) and let t be the negation of s. Then f(t) = 1 - f(s). ! Requirement 5: [f behaves right on existential quantifiers.] (a) Let s be a sentence in the domain of f of the form --- 614,625 ---- (b) If f(s) = 0, then for all a in A, f(a) = 0. ! Requirement 3: [f behaves right on negations.] Let s be a sentence in S \cap V(F(alpha) and let t be the negation of s. Then f(t) = 1 - f(s). ! Requirement 4: [f behaves right on existential quantifiers.] (a) Let s be a sentence in the domain of f of the form *************** *** 580,599 **** Then if f(s) = 1, then f(s_2) = 1. ! Requirement 6: [f behaves right on universal quantifiers]. - Let s be a sentence in the domain of f of the form - - "forall x psi(x)". - - Let s_1 be the sentence - - \neg exists x \neg psi(x) - - Then f(s) = f(s_1). Notice that s_1 will be in the domain of f. - - Requirement 7: [f behaves right on "=".] - (a). If s is is a sentence of the form "t = t" where t is a closed term, and s is in the domain of f, then f(s) = 1. --- 642,649 ---- Then if f(s) = 1, then f(s_2) = 1. ! Requirement 5: [f behaves right on "=".] (a). If s is is a sentence of the form "t = t" where t is a closed term, and s is in the domain of f, then f(s) = 1. *************** *** 612,619 **** 1, then f("t_2 \in t_4") = 1. (e) Let t_1, t_2 be closed terms in V(F(alpha)) and let n in ! omega. Then if f("t_1 = t_2") = 1, f("h_n(t_1) = h_n(t_2)") = 1. This completes our description of the tree T(X). Of course, we --- 662,671 ---- 1, then f("t_2 \in t_4") = 1. (e) Let t_1, t_2 be closed terms in V(F(alpha)) and let n in ! omega. [We suppose that h_n is a function symbol of our language ! L.] + Then if f("t_1 = t_2") = 1, f("h_n(t_1) = h_n(t_2)") = 1. This completes our description of the tree T(X). Of course, we *************** *** 656,665 **** As for compactness, obviously, if X has a model, then so does every bounded subset x of X which is in V_kappa. ! In fact, such a bounded subset will be in the domain of some ! member of the branch b that corresponds to the model [and which ! has rank > 0. But this member easily yields a model for x which is ! an element of V_kappa. Conversely, if every bounded subset of X has a model, level(alpha,T(X)) is non-empty for every alpha < kappa. Hence by --- 708,717 ---- As for compactness, obviously, if X has a model, then so does every bounded subset x of X which is in V_kappa. ! In fact, such a bounded subset will be in the domain of some ! member of the branch b that corresponds to the model [and ! which has rank some bfp > 0. But this member easily yields a ! model for x which is an element of V_kappa. Conversely, if every bounded subset of X has a model, level(alpha,T(X)) is non-empty for every alpha < kappa. Hence by *************** *** 706,714 **** 2. So on to our main topic. We have a model N of T_2 + "For no limit cardinal lambda is L_lambda a model of T_2". As I have remarked previously, it follows that there is a definable map ! from omega to OR, in N, whose range is cofinal in N. We let ! the map send i to gamma_i; we shall assume that the gamma_i's ! are lbfps and that the map i --> gamma_i is strictly increasing. There is no difficulty carrying out the definition of our --- 758,766 ---- 2. So on to our main topic. We have a model N of T_2 + "For no limit cardinal lambda is L_lambda a model of T_2". As I have remarked previously, it follows that there is a definable map ! from omega to OR, in N, whose range is cofinal in the ordinals ! of N. We let the map send i to gamma_i; we shall assume that ! the gamma_i's are lbfps and that the map i --> gamma_i is strictly increasing. There is no difficulty carrying out the definition of our *************** *** 752,758 **** Godel numbers [as Sigma_2 subsets of the class of sentences S] of X's that have models [in the sense that T(X) has a branch] is Pi^2. All that really matters is that this is a definable ! class in N. The precise hierarchy calculation does not matter. The compactness theorem goes through as before: X "has a model" [in the sense that T(X) has a branch] iff every subset x --- 804,810 ---- Godel numbers [as Sigma_2 subsets of the class of sentences S] of X's that have models [in the sense that T(X) has a branch] is Pi^2. All that really matters is that this is a definable ! class in N. The precise hierarchy calculation is unimportant. The compactness theorem goes through as before: X "has a model" [in the sense that T(X) has a branch] iff every subset x *************** *** 795,807 **** The main work is a length omega construction. One of the things that we will be building, step by step, is a series of ! Sigma_2 theories in our language L, T_0, T_1, ... This series of theories is increasing. Its union T* need not be Sigma_2, but it will certainly be a class of N. It will be evident from our construction that T* is complete. Moreover, it will be true for the limit theory that the xi_alpha's are ! indiscernibles. We will build the model M* according to the blueprint given by T* but using as generating indiscernibles the xi_n for n in Z --- 847,863 ---- The main work is a length omega construction. One of the things that we will be building, step by step, is a series of ! Sigma_2 theories T_0, T_1, ... [There is a slight conflict ! here with our previous use of T_2. I trust the context will ! disambiguate things.] + T_i will be in the language L_i. + This series of theories is increasing. Its union T* need not be Sigma_2, but it will certainly be a class of N. It will be evident from our construction that T* is complete. Moreover, it will be true for the limit theory that the xi_alpha's are ! indiscernibles. [T* will be in the language L_omega.] We will build the model M* according to the blueprint given by T* but using as generating indiscernibles the xi_n for n in Z *************** *** 837,847 **** (d) There are arbitrarily large lbfps. - (e) For no limit cardinal lambda is L_lambda a model of - Sigma_2-Replacement. - - [I think I don't actually need axiom 1(e), but it doesn't hurt.] - Axiom group 2: Axioms on the \bar{alpha}'s. For each alpha in OR, there is an axiom that says: --- 893,898 ---- *************** *** 864,918 **** xi_alpha \in xi_beta ! Axiom group 4: Axioms on the h_n's. ! (a) If x is not an ordinal, h_n(x) = 0. ! (b) If x is an ordinal, h_n(x) <= x. ! There is one instance of each axiom for each n in omega. - Axiom group 5: The "no new sets" axioms. ! [By the way, I believe these axioms are unnecessary. I am ! including them "to be safe".] - Let S be a set of sentences. We suppose that there is a finite - subset of omega, A, and a finite subset of OR, B such that: - - (a) if h_n appears in some s in S, then n \in A. - - (b) If xi_alpha appears in some s in S, then alpha \in B. - - - We have to construct some infinitary formulae. Lets start with - the following: - - If theta is a formula of L, then by theta^0 we shall mean - \neg theta . theta^1 will just mean theta. - - Now let f:S --> {0,1}. We introduce the formula A(S,f) which - roughly says that f correctly describes the truth values of - sentences in S: - - \bigand {theta^{f(theta)} | theta \in S} - - We can now formulate the instance of the "no new sets" axiom - corresponding to S, B(S): - - \bigvee {A(S,f) | f: S --> {0,1}}. - - So there is one such axiom in T_0 for each S that meets our - finiteness constraints. - - Axiom group 6: The least ordinal principle - The purpose of these axioms is to enforce, to the extent that we can, that the models of T_0 are well-founded. - [We definitely don't fully succeed. One can prove that the - ordinals of any model of T_0 are not well-ordered.] - Let X be a set of closed terms of L. There will be one instance of the axiom for each such X that meets the following finiteness constraint: --- 915,935 ---- xi_alpha \in xi_beta ! Axiom group 4: Axioms on the h_i's. ! (a) If x is not an ordinal, h_i(x) = 0. ! (b) If x is an ordinal, h_i(x) <= x. ! There are no instances of group 4 axioms in T_0. But if 0 < n, ! there will be the appropriate axioms of this type for each i < n. ! Axiom group 5: The least ordinal principle The purpose of these axioms is to enforce, to the extent that we can, that the models of T_0 are well-founded. Let X be a set of closed terms of L. There will be one instance of the axiom for each such X that meets the following finiteness constraint: *************** *** 920,928 **** There is a finite subset of omega, A, and a finite subset of OR, B such that: ! (a) if h_n appears in some x in X, then n \in A. ! (b) If xi_alpha appears in some x in X, then alpha \in B. Our axiom will have the form "If H(X) then C(X)". We describe these two components in turn: --- 937,945 ---- There is a finite subset of omega, A, and a finite subset of OR, B such that: ! (a) The h-support of any x in X is a subset of A. ! (b) The xi-support of any x in X is a subset of B. Our axiom will have the form "If H(X) then C(X)". We describe these two components in turn: *************** *** 981,997 **** up. I will return to the main thrust of the argument for Phase C in the next letter. - 1. In letter C3, I wrote: ! Uniformize S to a Sigma_2 function F with domain Y. ! ! But in that context, V_kappa does not have a canonical ! well-ordering so not all relations can be uniformized. ! ! We are, in fact, ok since the codomain of the relation in ! question consists of ordinals. ! ! 2. In stating my metatheory [ZF-] I remarked that it was not clear to me the relative consistency proofs I was giving could be carried out in primitive recursive arithmetic. --- 998,1005 ---- up. I will return to the main thrust of the argument for Phase C in the next letter. ! 1. In stating my metatheory [ZF-] I remarked that it was not clear to me the relative consistency proofs I was giving could be carried out in primitive recursive arithmetic. *************** *** 1001,1007 **** "Exponentiation is total". [This system is too weak to carry out the equiconsistency of ZF and GB.] ! 3. I've been somewhat sloppy about the formal details of our length omega constructions. [One occurred in phase B, and one will be about to be presented in phase C.] --- 1009,1015 ---- "Exponentiation is total". [This system is too weak to carry out the equiconsistency of ZF and GB.] ! 2. I've been somewhat sloppy about the formal details of our length omega constructions. [One occurred in phase B, and one will be about to be presented in phase C.] 24a25,26 > [But see my comments re this in letter C7.] > 32c34 < range is cofinal in N. We can and do assume that this map is --- > range is cofinal in OR. We can and do assume that this map is 54c56 < Jensen's original proof, and phase c. Thus had I chosen to --- > Jensen's original proof, and phase C. Thus had I chosen to 70,71c72,73 < lost that in the limit, though T will be of course, definable < in N. --- > will lose that in the limit, though T will be of course, > definable in N. 92c94 < rather like, but not exactly like the ones that figure in the --- > rather like, but not exactly like, the ones that figure in the 112a115,120 > Since writing the previous paragraph I have figured out how to > modify the usual Barise logic so as to apply to the V_kappa's > that we are considering in this series of lemmas. But the > point has no relevance to our current goals so I won't present > it. > 174,175c182,184 < 2. Our next task is to describe a certain infinitary < language. The finiteness condition we impose is non-standard --- > > 2. Our next task is to describe a certain family of infinitary > languages. The finiteness condition we impose is non-standard 180c189 < Basic predicates of the language: --- > There will be one language L_alpha for each alpha \leq omega. 181a191,192 > Basic predicates of the language L_alpha: > 186c197 < For each n in omega, there is a unary function h_n. These will --- > For each n < alpha, there is a unary function h_n. These will 189,191c200 < Variables: It seems to me that I can get by with just < countably many. But to be on the safe side, I will have a < variable v_alpha for each ordinal alpha. --- > Variables: 192a202,204 > There will be a variable v_alpha for each ordinal alpha. > > 220a233,244 > Some comments: Barwise works with Skolem functions rather than > Henkin constants. Normally that would be the right thing to > do. But for our purposes, the use of Henkin constants proves > more convenient. > > We define the xi-support of a term or formula to be the set of > alpha such that xi_alpha appears in the term or formula. > > The notion of the h-support of a term or formula is > analogous. It is the set of n such that h_n appears in the > term or formula. > 228,229c252,253 < (a) if psi is a formula and x is a variable, then \forall x < psi and \exists x psi are formulas. --- > (a) if psi is a formula and x is a variable, then \exists x > psi is a formula. 247,249d270 < dually, If S is a set of formulas, then <\bigand, S> is a < formula [and denotes the the infinite conjunction of the < formulas in S]. 274a296,302 > We have chosen not to have \forall and infinite conjunctions > to be an official part of our infinitary languages. However, > they can be introduced by abbreviations [inspired by de > Morgan's laws] in the usual way. We will feel free to use them > when informally describing formulas. > > 287a316,348 > I am next going to explicitly spell out what set corresponds > to each term or formula. The definition is inductive; we let > <> be the set corresponding to the term or formula s. > > The definition that follows is totally routine. There is > nothing tricky going on. > > <> = <0,alpha>; > > <> = <1,alpha>; > > <<\bar{alpha}>> = <2, alpha>; > > << \iota x psi >> = <3, <>, <> >; > > << h_n (t) >> = <4, n, <> >; > > << \exists x psi >> = <5, <>, <> >; > > << \neg psi >> = <6, <> > ; > > Now let S be a set of formulas. Let S* be {<> | s \in S}. > > << \bigvee S >> = <7, S* > ; > > << s \in t >> = <8, <>, <> > > > << s = t >> = <9, <>, <> > > > I don't think I will ever use explicitly the definition just > given. But it is already being implicitly used in the > immediately following paragraph. > 322a384,385 > > 374c437 < sens alpha to level(alpha, T) is Delta_2. --- > sends alpha to level(alpha, T) is Delta_2. 419,420c482,483 < ZC + Sigma_2-replacement proves Sigma_2-uniformization. < [I. e., there is no need to require V=L.] --- > ZC + Sigma_2-replacement proves the following special case of > Sigma_2-uniformization. [I. e., there is no need to require V=L.] 421a485,494 > Let R(x,y) be a Sigma_2-relation. We suppose that whenever > R(a,b) obtains then b is an ordinal. Then there is a > Sigma_2-relation S such that: > > (1) If S(a,b), then R(a,b). > > (2) If S(a,b) and S(a,b'), then b = b'. > > (3) If R(a,b), then for some b', S(a,b'). > 480,482c553,555 < We introduced an infinitary language L in letter c2. Recall < that S is the set of sentences of L. [From the standpoint of < V_kappa, S is a proper class.] Let X be a Sigma_2 subset of --- > Let L be one of the languages L_alpha introduced in letter > C2. Let S be the set of sentences of L. [From the standpoint > of V_kappa, S is a proper class.] Let X be a Sigma_2 subset of 533c606 < Requirement 2: [f behaves right on conjunctions]. --- > Requirement 2: [f behaves right on disjunctions.] 535,545d607 < Suppose that s is the conjunction of the set of sentences A. < Suppose also that s \in S and s \in V(F(alpha)). Notice < that this entails that each member of A is in S \cap < V(F(alpha)), < < (a) If f(s) = 1, then for all a in A, f(a) = 1. < < (b) If f(s) = 0, then for some a in A, f(a) = 0. < < Requirement 3: [f behaves right on disjunctions.] < 555c617 < Requirement 4: [f behaves right on negations.] --- > Requirement 3: [f behaves right on negations.] 560c622 < Requirement 5: [f behaves right on existential quantifiers.] --- > Requirement 4: [f behaves right on existential quantifiers.] 583c645 < Requirement 6: [f behaves right on universal quantifiers]. --- > Requirement 5: [f behaves right on "=".] 585,596d646 < Let s be a sentence in the domain of f of the form < < "forall x psi(x)". < < Let s_1 be the sentence < < \neg exists x \neg psi(x) < < Then f(s) = f(s_1). Notice that s_1 will be in the domain of f. < < Requirement 7: [f behaves right on "=".] < 615c665,666 < omega. --- > omega. [We suppose that h_n is a function symbol of our language > L.] 616a668 > 659,662c711,714 < In fact, such a bounded subset will be in the domain of some < member of the branch b that corresponds to the model [and which < has rank > 0. But this member easily yields a model for x which is < an element of V_kappa. --- > In fact, such a bounded subset will be in the domain of some > member of the branch b that corresponds to the model [and > which has rank some bfp > 0. But this member easily yields a > model for x which is an element of V_kappa. 709,711c761,763 < from omega to OR, in N, whose range is cofinal in N. We let < the map send i to gamma_i; we shall assume that the gamma_i's < are lbfps and that the map i --> gamma_i is strictly --- > from omega to OR, in N, whose range is cofinal in the ordinals > of N. We let the map send i to gamma_i; we shall assume that > the gamma_i's are lbfps and that the map i --> gamma_i is strictly 755c807 < class in N. The precise hierarchy calculation does not matter. --- > class in N. The precise hierarchy calculation is unimportant. 798c850,852 < Sigma_2 theories in our language L, T_0, T_1, ... --- > Sigma_2 theories T_0, T_1, ... [There is a slight conflict > here with our previous use of T_2. I trust the context will > disambiguate things.] 799a854,855 > T_i will be in the language L_i. > 804c860 < indiscernibles. --- > indiscernibles. [T* will be in the language L_omega.] 840,844d895 < (e) For no limit cardinal lambda is L_lambda a model of < Sigma_2-Replacement. < < [I think I don't actually need axiom 1(e), but it doesn't hurt.] < 867c918 < Axiom group 4: Axioms on the h_n's. --- > Axiom group 4: Axioms on the h_i's. 869c920 < (a) If x is not an ordinal, h_n(x) = 0. --- > (a) If x is not an ordinal, h_i(x) = 0. 871c922 < (b) If x is an ordinal, h_n(x) <= x. --- > (b) If x is an ordinal, h_i(x) <= x. 873c924,925 < There is one instance of each axiom for each n in omega. --- > There are no instances of group 4 axioms in T_0. But if 0 < n, > there will be the appropriate axioms of this type for each i < n. 875d926 < Axiom group 5: The "no new sets" axioms. 877,878c928 < [By the way, I believe these axioms are unnecessary. I am < including them "to be safe".] --- > Axiom group 5: The least ordinal principle 880,909d929 < Let S be a set of sentences. We suppose that there is a finite < subset of omega, A, and a finite subset of OR, B such that: < < (a) if h_n appears in some s in S, then n \in A. < < (b) If xi_alpha appears in some s in S, then alpha \in B. < < < We have to construct some infinitary formulae. Lets start with < the following: < < If theta is a formula of L, then by theta^0 we shall mean < \neg theta . theta^1 will just mean theta. < < Now let f:S --> {0,1}. We introduce the formula A(S,f) which < roughly says that f correctly describes the truth values of < sentences in S: < < \bigand {theta^{f(theta)} | theta \in S} < < We can now formulate the instance of the "no new sets" axiom < corresponding to S, B(S): < < \bigvee {A(S,f) | f: S --> {0,1}}. < < So there is one such axiom in T_0 for each S that meets our < finiteness constraints. < < Axiom group 6: The least ordinal principle < 913,915d932 < [We definitely don't fully succeed. One can prove that the < ordinals of any model of T_0 are not well-ordered.] < 923c940 < (a) if h_n appears in some x in X, then n \in A. --- > (a) The h-support of any x in X is a subset of A. 925c942 < (b) If xi_alpha appears in some x in X, then alpha \in B. --- > (b) The xi-support of any x in X is a subset of B. 984d1000 < 1. In letter C3, I wrote: 986,994c1002 < Uniformize S to a Sigma_2 function F with domain Y. < < But in that context, V_kappa does not have a canonical < well-ordering so not all relations can be uniformized. < < We are, in fact, ok since the codomain of the relation in < question consists of ordinals. < < 2. In stating my metatheory [ZF-] I remarked that it was not --- > 1. In stating my metatheory [ZF-] I remarked that it was not 1004c1012 < 3. I've been somewhat sloppy about the formal details of our --- > 2. I've been somewhat sloppy about the formal details of our From solovay@ccrwest.org Tue Mar 6 16:27:04 2001 Date: Tue, 6 Mar 2001 16:27:03 -0800 (PST) From: "Robert M. Solovay" To: Randall Holmes Subject: Phase C: first letter Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: Randall, 1. Here starts the series of letters in which I present the proof that Con(ZC + Sigma_2 Replacement) entails Con(NFU*). The usual remarks about the metatheory apply. Officially, my metatheory is ZF-. But the proof would easily transcribe to Peano Arithmetic. I suspect that it would go through in PRA as well, but i definitely have not thought through the details that would establish this last claim. [But see my comments re this in letter C7.] We are given a model N of ZC + Sigma_2 Replacement, N. We can and do assume that V=L holds in N. Moreover, we can and do assume that for no limit cardinal lambda of N is L_lambda a model of Sigma_2 replacement. It follows, as I discussed in the first letter of phase B, that there is a definable map from omega to OR, in N, whose range is cofinal in OR. We can and do assume that this map is strictly increasing, and that its range consists of lbfp's. 2. There is much in common between the proof in phase C and the earlier proof in phase B, and I shall begin by reviewing these comment elements. Before doing that, I make the following remark: There is a clear lineage to this proof that runs as follows: Jensen's proof that for every ordinal alpha, there is an alpha model of NFU*. Phase A. Phase B. Phase C. Each phase has a lot in common with its immediate predecessor. But there is very little commonality [though there is some-- for example, the use of Erdos-Rado] between Jensen's original proof, and phase C. Thus had I chosen to present phase C without indicating its origins, it would look much more original than it actually is. 2. Here is a very high level outline of the proof. Again, we will be constructing a certain theory T [that plays the role of the "term model" M of phase B.]. The construction will take place in N, and proceed in omega stages. Roughly speaking, this has the effect that we are dealing with only a "set's worth" of problems at any stage. Because there is a definable cofinal map of omega into OR, we will succeed in "handling everything" by the end of our construction. We remark that the theory T will be a proper class of the model N. Indeed, even the approximations to T [call them T_n] will be proper classes of N. The T_n's will be Sigma_2, but we will lose that in the limit, though T will be of course, definable in N. [I haven't bothered to compute what its complexity is. My guess is that it is either Sigma_3 or Sigma_4.] The theory T will have a canonical term-model M*. M* will be a model of a significant amount of set-theory: roughly, MacLane set-theory + KP + V=L + "There are arbitrarily large lbfps." The model M* will have an obvious automorphism and so yield, in the usual way, a model Q of NFU. We will take steps during our construction to ensure that the W [strongly cantorian initial segment of Z] of the model Q is canonically identified with the model N. This together with the fact that Q is a definable class of N will ensure, in the usual way, that Q is a model of NFU*. 3. Next let me emphasize what is different in the proof. Rather than working with a term language as I did in phases A and B I will be working in a certain infinitary language, rather like, but not exactly like, the ones that figure in the Barwise compactness theorem. It turns out [and I only learned of this this past Sunday] that if V_kappa is a model ZC + Sigma_2 replacement and kappa has cofinality omega, then the infinitary logic based on V_kappa has strong compactness properties. [I will review this material in a subsequent letter.] One can apply these results "inside N". The "proofs" are sets of N, while the models constructed are proper classes of N. By using this machinery, we are able to carry out an analogue of the proof given in phase B for the weaker theory T_2. One thing that is peculiar about these results for V_kappa: The proofs of formal logic bear some resemblance to informal proofs. And the proofs involved in the usual Barwise compactness theorem bear a passing resemblance to formal proofs in first-order logic. But the "proofs" of the V_kappa framework don't look like proofs at all. Since writing the previous paragraph I have figured out how to modify the usual Barise logic so as to apply to the V_kappa's that we are considering in this series of lemmas. But the point has no relevance to our current goals so I won't present it. This ends letter 1 of phase C. My next task is to present my version of the results about V_kappa. [Although it is all in Barwise's book "Admissible sets and structures", it took me a while to decode what he was saying. And when I did, I realized that I would present the material quite differently than he did.] From solovay@ccrwest.org Tue Mar 6 22:35:15 2001 Date: Tue, 6 Mar 2001 22:35:15 -0800 (PST) From: "Robert M. Solovay" To: Randall Holmes Subject: Letter 2 of phase C Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: This letter is devoted to my exposition of the theory of compactness for uncountable structures V_kappa where kappa has cofinality omega. My source for all this is Barwise's book Admissible Sets and Structures". Particularly relevant is section 7 of Chapter VIII of Barwise. Especially Theorems 7.2 and 7.4. That said, I found Barwise's exposition of this needlessly long-winded and obscure. I hope to do a better job. 1. We will be working in ZFC. Our main focus is the following situation: (a) kappa is a cardinal of cofinality omega; (b) V_kappa is a model of ZC + Sigma_2-Replacement. As Barwise remarks, the arguments will go through in the following more general situation: (a) kappa is a cardinal of cofinality omega; (b) S is a subset of V_kappa; (c) P(x,y) is the relation : y is the power set of x; (d) define the notion of Delta_0 formula in the usual way except we allow atomic formulas of the form Sx and P(x,y) as well as those of the form x=y and x epsilon y. (e) The structure is admissible, Our special case corresponds to taking S to be the empty set. To get the general case, replace Sigma_2 by "Sigma_1 in the predicates P and S". 2. Our next task is to describe a certain family of infinitary languages. The finiteness condition we impose is non-standard and is needed for our particular application. The general results I review in this letter would go through just as well if the finiteness condition were not imposed. There will be one language L_alpha for each alpha \leq omega. Basic predicates of the language L_alpha: Exactly as in set-theory, there are two: = and \epsilon. Functions: For each n < alpha, there is a unary function h_n. These will play the same role as in the previous phases of the proof. Variables: There will be a variable v_alpha for each ordinal alpha. Constants: These come in two flavors: 1) For each ordinal alpha [less than kappa-- we are working within V_kappa!] there will be a constant \bar{alpha}. The intended interpretation of \bar{alpha} is alpha. 2) For each ordinal alpha, there will be a constant xi_alpha. These play a somewhat analogous role to the xi_i's in the previous phases of the proof. And at a later point, I will need also xi_i's for negative integers i. But they won't appear in the current language definition. The definition of the language is inductive. In particular, there will be Henkin constants introduced by the following rule; If psi(x) is a formula having only the variable x free, then \iota x psi(x) is a constant of the language. Subsequently, I will need the notion of, for example, a xi_alpha appearing in some term or formula. One gives the obvious inductive definition. But in particular, if xi_alpha appears in the formula psi(x), it appears in the Henkin constant \iota x psi (x). Some comments: Barwise works with Skolem functions rather than Henkin constants. Normally that would be the right thing to do. But for our purposes, the use of Henkin constants proves more convenient. We define the xi-support of a term or formula to be the set of alpha such that xi_alpha appears in the term or formula. The notion of the h-support of a term or formula is analogous. It is the set of n such that h_n appears in the term or formula. The definition of term is evident as is that of atomic formula. [Though note well, that because of our use of Henkin constants, the notion of term and formula must really be given a simultaneous inductive definition.] The other clauses of the definition are: (a) if psi is a formula and x is a variable, then \exists x psi is a formula. (b) if psi is a formula, so is \neg psi. (c) The final clause is a little more involved. We give first the usual version that is found in Barwise, and then the amendment that we will actually use. Imagine that \bigvee is the kind of "big V" used to denote a possibly infinite disjunction. And imagine that \bigand is a large upside down V of the sort used to denote a possibly infinite conjunction. The approximate version of clause (c) is: If S is a set of formulas, then <\bigvee, S> is a formula [and denotes the infinite disjunction of the formulas in S]; CAUTION: When we say "set" here we really mean element of V_kappa. To get our precise definition, we need to impose three finiteness conditions on S. [The first is imposed by Barwise as well.] Finiteness conditions: 1) The set of alpha such that v_alpha occurs free in some formula of S is finite. 2) The set of alpha such that xi_alpha occurs in some formula of S is finite. 3) The set of n such that h_n appears in some formula of S is finite. Note well; We do not require that the set of alpha such that \bar{alpha} appears in some formula of S is finite. This completes our description of the syntax of our infinitary language L. We have chosen not to have \forall and infinite conjunctions to be an official part of our infinitary languages. However, they can be introduced by abbreviations [inspired by de Morgan's laws] in the usual way. We will feel free to use them when informally describing formulas. I hope the semantics of L is evident. In particular, given a sentence of L [formula with no free variables] and an L-structure [a structure where all the constants and functions as well as the epsilon predicate are given interpretations], then the sentence has a truth-value relative to this structure which is defined in the evident way. Notice that the way we have set things up, the sentences, formulas, terms, etc, are elements of V_kappa. However, it will be important to allow structures which ar not elements of V_kappa. [We could get by considering just structures whose underlying sets are subsets of V_kappa.] I am next going to explicitly spell out what set corresponds to each term or formula. The definition is inductive; we let <> be the set corresponding to the term or formula s. The definition that follows is totally routine. There is nothing tricky going on. <> = <0,alpha>; <> = <1,alpha>; <<\bar{alpha}>> = <2, alpha>; << \iota x psi >> = <3, <>, <> >; << h_n (t) >> = <4, n, <> >; << \exists x psi >> = <5, <>, <> >; << \neg psi >> = <6, <> > ; Now let S be a set of formulas. Let S* be {<> | s \in S}. << \bigvee S >> = <7, S* > ; << s \in t >> = <8, <>, <> > << s = t >> = <9, <>, <> > I don't think I will ever use explicitly the definition just given. But it is already being implicitly used in the immediately following paragraph. It is fairly easy to see that the notions of formula, term, etc. are Delta_2(V_kappa). We Godel number the Sigma_2 subsets of V_kappa in an evident way. [I had to do essentially this when I Godel numbered Sigma_2 partial functions in letter 3 of phase B.] Let S be the set of sentences of L. [S is a "proper class' from the standpoint of V_kappa.] We get a Godel numbering of the Sigma_2 subsets of S as follows. The subset of S with Godel number e is obtained by intersecting S with the subset of V_kappa with Godel number e. [e here is some element of V_kappa.] Theorem: Sigma_2 completeness. The set of e such that the collection of sentences with Godel number e has a model [not necessarily in V_kappa!] is Pi_2. Theorem: Sigma_2 compactness. Let A be a Sigma_2 set of sentences of L. Then the following are equivalent: 1) A has a model. 2) For every a \subseteq A such that a \in V_kappa, a has a model. 3) For every a \subseteq A such that a \in V_kappa, a has a model which is an element of V_kappa. This completes this letter. These theorems are not at all evident. I will prove them in the next letter [or two]. From solovay@ccrwest.org Wed Mar 7 12:50:07 2001 Date: Wed, 7 Mar 2001 12:50:06 -0800 (PST) From: "Robert M. Solovay" To: Randall Holmes Subject: C: letter 3 Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: Randall, Our current order of business is to prove the facts concerning infinitary logic that I asserted at the end of letter C2. Our approach will be as follows. 1) We shall see that a version of Konig's lemma holds in our current context [with "the usual proof"]. 2) It is well-known that the usual form of Konig's lemma is closely related to the Godel completeness theorem. A similar situation will obtain here and we shall prove the theorems on completeness and compactness asserted at the end of letter C2 from the current version of Konig's lemma. 2. Let's rehearse some well-known definitions. A tree T is a set |T| together with a partial-ordering, \leq_T such that the predecessors of any x in T are well-ordered by \leq_T. To avoid any ambiguity: \leq_T is reflexive. Let T be a tree. Each x in |T| has an ordinal attached, its level, which is the order-type of the set of y which are strictly less than x in the tree. I write level(alpha,T) for the set of points of T which appear at level alpha. 3. Next I have to explain when a tree T is "a Delta_2 tree". We require: 1) The underlying set of T is a Delta_2 subset of V_kappa. [So from the standpoint of V_kappa, T may be a proper class. 2) The partial-ordering of T is Delta_2. 3) The height of T is at most kappa. For each alpha < kappa, level(alpha, T) is an element of V_kappa. And the map which sends alpha to level(alpha, T) is Delta_2. 4. We can now formulate the version of Konig's lemma we will be proving. But first, I should probably recall our standing assumptions on kappa. 1) kappa is a limit cardinal of cofinality omega; 2) V_kappa is a model of ZC + Sigma_2-replacement. Here is the version of Konig's lemma we will be proving. Lemma 4. Let T be a Delta_2 tree. Suppose that for every alpha < kappa, level(alpha, T) is non-empty. Then T has a branch of order-type kappa. [A branch is a maximal linearly ordered subset of T.] Cf. Barwise op. cit. Chapter VIII, Theorem 7.2. Lemma 4 will follow immediately from the fact that kappa has cofinality omega and the following two claims; Claim 1. Let alpha < kappa. Then there is an x at level alpha in T such that for every beta > alpha, [with beta < kappa, of course] x has a descendent at level beta in T. Claim 2. Let x in T. We suppose that x has level alpha and that for every beta with alpha < beta < kappa, x has a descendent at level beta of the tree. Let beta_0 be given with alpha < beta_0 < kappa. Then x has a descendent y at level beta_0 with the following property: For every gamma with beta_0 < gamma < kappa, y has a descendent at level gamma in T. [When I say, e,g,. that x has a descendent y at level beta in T, I mean there is a y at level beta such that x \leq_T y.] The proofs of 1 and 2 are quite similar and I shall content myself with a proof of 2. But before beginning the proof, let me make a remark that I should have made earlier: ZC + Sigma_2-replacement proves the following special case of Sigma_2-uniformization. [I. e., there is no need to require V=L.] Let R(x,y) be a Sigma_2-relation. We suppose that whenever R(a,b) obtains then b is an ordinal. Then there is a Sigma_2-relation S such that: (1) If S(a,b), then R(a,b). (2) If S(a,b) and S(a,b'), then b = b'. (3) If R(a,b), then for some b', S(a,b'). The proof uses the fact that ZC + Sigma_2-Replacement proves: (a) For every alpha, V_alpha exists: (b) For every alpha, there is bfp greater than alpha. (c) [Levy absoluteness] If alpha is a bfp, V_alpha is absolute for Sigma_1 sentences. Let's begin the proof of claim 2. We are given x satisfying the hypotheses of claim 2. Towards a contradiction, assume that the conclusion of the lemma fails for some beta_0 > alpha. Let Y be the set of descendants of x at level beta_0. By assumption Y is non-empty. Introduce a Sigma_2 relation S(a,b) thus; a is in Y, b is an ordinal greater than beta_0 and a has no descendants at level b. Our assumptions imply that for every a in Y, there is a b such that S(a,b). Uniformize S to a Sigma_2 function F with domain Y. Let gamma be an ordinal greater than every ordinal in the range of F. By assumption, x has a descendant at level gamma, say z. Let y be the ancestor of z at level beta_0. Then y in Y, and gamma > F(y). But this contradicts the definition of F since y has the descendant z at level gamma. This completes our proof of the variant of Konig's Lemma and with it letter C3. From solovay@ccrwest.org Wed Mar 7 18:01:01 2001 Date: Wed, 7 Mar 2001 18:01:01 -0800 (PST) From: "Robert M. Solovay" To: Randall Holmes Subject: Letter C4 Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: The goal of this letter is to use the variant of Konig's Lemma proved in letter C3 to establish the theorems asserted at the end of letter C2. 1. Let L be one of the languages L_alpha introduced in letter C2. Let S be the set of sentences of L. [From the standpoint of V_kappa, S is a proper class.] Let X be a Sigma_2 subset of S. We are going to associate to X a certain tree T(X). Roughly speaking, models of X will correspond to branches of T(X). T(X) will be a subtree of a certain universal tree, T*, which we build first. Work in V_kappa. We define a Delta_2 function, F, mapping kappa to kappa as follows: F(0) = 0. The ordinals F(1), F(2), ... enumerate in increasing order the lbfps less than kappa. [Implicit in this definition is the fact that there are kappa lbfps less than kappa.] We now describe the tree T*. The tree order will be inclusion. Level 0 of the tree will consist of one member, the empty set. Let alpha > 0. Let S_alpha consist of those sentences of L which lie in V(F(alpha)). Then the alpha^{th} level of T* consists of all maps of S_alpha into (0,1}. This completes the definition of T*. 2. We follow the usual conventions of thinking of 1 as "truth" and 0 as "falsehood". Very roughly, level alpha of T(X) will consist of those elements of level alpha of T* which "respect the logical connectives" and require those elements known to be in X at stage F(alpha) to get truth value 1. We turn to the precise requirements. So let alpha be given and let f be in level alpha of T*. We describe precisely the requirements that f must satisfy to be in level alpha of T(X). If alpha = 0, then the empty set will be in the bottom level of T(X). So assume from now on, that alpha > 0. Let the Sigma^2 definition of X have the form (exists x)A(x,s) where A is Pi_1. Requirement 1: Suppose that s \in V(F(alpha), s is a sentence, and for some x in V(F(alpha)) A(x,s). Then f(s) = 1. Requirement 2: [f behaves right on disjunctions.] Suppose that s is the disjunction of the set of sentences A. Suppose also that s \in S and s \in V(F(alpha)). Notice that this entails that each member of A is in S \cap V(F(alpha)), (a) If f(s) = 1, then for some a in A, f(a) = 1. (b) If f(s) = 0, then for all a in A, f(a) = 0. Requirement 3: [f behaves right on negations.] Let s be a sentence in S \cap V(F(alpha) and let t be the negation of s. Then f(t) = 1 - f(s). Requirement 4: [f behaves right on existential quantifiers.] (a) Let s be a sentence in the domain of f of the form "exists x psi(x)" Let t be a closed term of L lying in V(F(alpha)). Let s_1 be the sentence psi(t). Then if f(s_1) = 1, f(s) = 1. (b) Let s be a sentence in the domain of f of the form "exists x psi(x)". Let t_1 be the closed term \iota x psi(x) and let s_2 be the sentence "psi(t_1)". Then if f(s) = 1, then f(s_2) = 1. Requirement 5: [f behaves right on "=".] (a). If s is is a sentence of the form "t = t" where t is a closed term, and s is in the domain of f, then f(s) = 1. (b) Suppose that t_1 and t_2 are closed terms in V(F(alpha)). If f("t_1 = t_2") = 1, then f("t_2 = t_1") = 1. (c) Suppose that t_1, t_2, and t_3 are closed terms in V(F(alpha)). Then if f("t_1 = t_2") = 1, and f("t_2 = t_3") = 1, then f("t_1 = t_3") = 1. (d) Suppose that t_1, t_2, t_3, t_4 are closed terms in V(F(alpha)). If f("t_1 = t_2") = 1, f("t_3 = t_4") = 1, and f("t_1 \in t_3") = 1, then f("t_2 \in t_4") = 1. (e) Let t_1, t_2 be closed terms in V(F(alpha)) and let n in omega. [We suppose that h_n is a function symbol of our language L.] Then if f("t_1 = t_2") = 1, f("h_n(t_1) = h_n(t_2)") = 1. This completes our description of the tree T(X). Of course, we have just "done the obvious thing". It is evident that T(X) is a Delta_2 tree in the sense of letter C3. 3. Recall that a branch through a tree such as T(X) is a maximal linearly ordered subset that has order type kappa. So it has elements on every level. So if b is a branch through T(X), the union of b is a function mapping S to {0,1} that gives each s in X the value 1 and respects the various logical connectives. If M is a model of X, then M determines a function h:S --> {0,1} ["compute the truth value of the sentence s in M"]. The restriction of h to the various S_alpha's gives a branch through T(X). Conversely, if b is a branch through T(X), b determines a model of X as follows. First, we put an equivalence relation on the closed terms by saying t_1 == t_2 iff the union of b gives "t_1 = t_2" value 1. The underlying set of M is the set of equivalence classes. We define the other elements of the structure of M "according to b" in an evident way. Because of the requirements imposed on T(X), it is routine to check that a sentence is true in M iff the union of b gives it the value 1. [The fact that L is Henkenized is key here, of course.] The various claims at the end of letter C2 are now pretty evident. First, X has a model iff T(X) has a branch iff every level of T(X) is non-empty. But this last formulation is clearly Pi_2. As for compactness, obviously, if X has a model, then so does every bounded subset x of X which is in V_kappa. In fact, such a bounded subset will be in the domain of some member of the branch b that corresponds to the model [and which has rank some bfp > 0. But this member easily yields a model for x which is an element of V_kappa. Conversely, if every bounded subset of X has a model, level(alpha,T(X)) is non-empty for every alpha < kappa. Hence by Konig, T(X) has a branch, so X has a model. The various claims made at the end of letter C2 have been proved. This ends letter C4. From solovay@ccrwest.org Wed Mar 7 22:54:39 2001 Date: Wed, 7 Mar 2001 22:54:39 -0800 (PST) From: "Robert M. Solovay" To: Randall Holmes Subject: Letter C5 Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: The main purpose of this [hopefully brief] letter is to discuss what the compactness and completeness results proved in the last two letters look like when formulated "internal to our model N". But first I want to correct an unimportant misstatement that I made. I said in letter C2, as I recall, that the "proofs" for the compactness results formulated for V_kappa in C_2 look nothing like those for the usual Barwise compactness. But in fact they can be made to look quite similar. One basically has one new type of axiom saying that if S is a set of sentences, then any truth assignment to S lies in the ground model. [This can be expressed as a big infinitary disjunction.] In our current context, this is a little awkward to carry out since the new axioms don't meet our finiteness restraints. This difficulty can be circumvented, but since there is no particular need to do so I shall say no more about this. 2. So on to our main topic. We have a model N of T_2 + "For no limit cardinal lambda is L_lambda a model of T_2". As I have remarked previously, it follows that there is a definable map from omega to OR, in N, whose range is cofinal in the ordinals of N. We let the map send i to gamma_i; we shall assume that the gamma_i's are lbfps and that the map i --> gamma_i is strictly increasing. There is no difficulty carrying out the definition of our language L internal to N. Where we previously had, for example, a constant \bar{alpha} for each alpha < kappa, we now have such a constant for each ordinal alpha. The collections of terms, sentences, etc. are proper classes which are Delta_2. If X is a Sigma_2 set of sentences of L, we can form the tree T(X) much as before. It will once again be a Delta_2 tree. Konig's lemma now takes the following form: If every level of T(X) is non-empty then there is a definable branch through T. The old proof goes through, mutatis mutandis, using in a crucial way that there is a definable map that shows that OR has cofinality omega. Such a branch through T(X) yields a canonical model M of X. I will make one small change in the definition of M. Each equivalence class of closed terms has a least member with respect to the canonical well-ordering of N. [Recall that N is a model of V=L.] I will take these least members rather than the equivalence classes themselves as the elements of the underlying class of M. [In general, M will be given as a proper class of the model N.] If one is being truly pedantic, there are proper classes which give; (a) the underlying class of M; (b) The map from the class of closed terms of L onto M; (c) The class of sentences of L which are true. This gives us a quite satisfactory grip on M. The completeness theorem holds in the form that the class of Godel numbers [as Sigma_2 subsets of the class of sentences S] of X's that have models [in the sense that T(X) has a branch] is Pi^2. All that really matters is that this is a definable class in N. The precise hierarchy calculation is unimportant. The compactness theorem goes through as before: X "has a model" [in the sense that T(X) has a branch] iff every subset x of X has a set-model. There is one small point in the proof of this that I slurred over in the last letter. Let x be a subset of X. Then using Sigma_2-replacement one sees that there is some stage alpha [which is an lbfp] at which all the Sigma_2 facts that show each member of x actually is in X have witnesses in V(alpha). This is needed to get a model of x from an appropriate node of the tree T(X). This completes my discussion of the problem of internalizing the results of Letters C3 and C4. End of letter C5. From solovay@ccrwest.org Thu Mar 8 12:38:43 2001 Date: Thu, 8 Mar 2001 12:38:42 -0800 (PST) From: "Robert M. Solovay" To: Randall Holmes Subject: Letter C6 Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: This letter starts the main thrust of the argument. Let me outline how it will go. We have our model N of T_2 whose properties I have just recalled in letter C5. Our construction will take place "inside N". The main work is a length omega construction. One of the things that we will be building, step by step, is a series of Sigma_2 theories T_0, T_1, ... [There is a slight conflict here with our previous use of T_2. I trust the context will disambiguate things.] T_i will be in the language L_i. This series of theories is increasing. Its union T* need not be Sigma_2, but it will certainly be a class of N. It will be evident from our construction that T* is complete. Moreover, it will be true for the limit theory that the xi_alpha's are indiscernibles. [T* will be in the language L_omega.] We will build the model M* according to the blueprint given by T* but using as generating indiscernibles the xi_n for n in Z [Here Z is the set of all integers both positive and negative as computed in N.] M* has an evident automorphism which sends xi_i to xi_{i+1} for any i. M* will be a model of a moderately strong set-theory and the xi_n's will be lbfps in that model, so we will get a model Q of NFU in the usual way. It will be evident from our construction that the strongly cantorian ordinals of Q are canonically identified with the ordinals of N. In the usual way, this plus the fact that Q is given by a class of N will ensure that the crucial axioms of counting and strongly Cantorian full selection hold in Q so that Q is indeed a model of NFU*. 2. The next order of business is to describe the theory T_0 and prove it has a model. [I always mean by this that the relevant tree T(T_0) has a branch.] Most of the axioms of T_0 are routine, but there is at least one tricky one that I will call attention to when we get to it. Axiom group 1: Axioms of set-theory We can find finitely many axioms in the usual language of set-theory [first-order logic is a subset of L!] that express: (a) Maclane set-theory; (b) KP; (c) V=L; (d) There are arbitrarily large lbfps. Axiom group 2: Axioms on the \bar{alpha}'s. For each alpha in OR, there is an axiom that says: For all x [x is in \bar{alpha} iff \bigvee {x = \bar{beta} | beta < alpha} [This is the usual way one pins down the meaning of \bar{alpha} in infinitary logic.] Axiom group 3: Axioms on the xi_alpha's. For each ordinal alpha, there is an axiom: xi_alpha is an lbfp. For each pair of ordinals, alpha, beta with alpha < beta, there is an axiom: xi_alpha \in xi_beta Axiom group 4: Axioms on the h_i's. (a) If x is not an ordinal, h_i(x) = 0. (b) If x is an ordinal, h_i(x) <= x. There are no instances of group 4 axioms in T_0. But if 0 < n, there will be the appropriate axioms of this type for each i < n. Axiom group 5: The least ordinal principle The purpose of these axioms is to enforce, to the extent that we can, that the models of T_0 are well-founded. Let X be a set of closed terms of L. There will be one instance of the axiom for each such X that meets the following finiteness constraint: There is a finite subset of omega, A, and a finite subset of OR, B such that: (a) The h-support of any x in X is a subset of A. (b) The xi-support of any x in X is a subset of B. Our axiom will have the form "If H(X) then C(X)". We describe these two components in turn: H(X) will express that all the members of x are ordinals: \bigand {x is an ordinal | x \in X} C(X) will express that some member of X is least: To start, let [for x in X], D(x,X) express that x is least in X: \bigand { x <= y | y in X} Then C(X) is just the obvious infinite disjunction: \bigvee {D(x,X) | x \in X} This completes our description of the theory T_0. It is obviously Sigma^2. In fact, it is obviously Delta_2. I say that T_0 is consistent [in the usual meaning we have been giving to such phrases that the corresponding tree T(T_0) has a branch through it]. For this, its enough, by compactness, to check that every set of axioms of T_0 has a model. But this is easy. For any set, a, of axioms of T_0, it is easy to whip up a model whose underlying set is L_lambda, where lambda = F(gamma^+). Here F is the function from letter C4 that enumerates the lbfps; gamma^+ is the least cardinal greater than gamma; and gamma \in OR is chosen sufficiently large compared to a. This is a good place to pause and I will end letter C6 here. From solovay@ccrwest.org Thu Mar 8 14:37:43 2001 Date: Thu, 8 Mar 2001 14:37:42 -0800 (PST) From: "Robert M. Solovay" To: Randall Holmes Subject: Letter C7 Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: Randall, This letter handles a number of minor points that have come up. I will return to the main thrust of the argument for Phase C in the next letter. 1. In stating my metatheory [ZF-] I remarked that it was not clear to me the relative consistency proofs I was giving could be carried out in primitive recursive arithmetic. While I still am reluctant to make an official claim, I've thought about things some more, and it looks routine to carry out the proofs in even the much weaker system IDelta_0 + "Exponentiation is total". [This system is too weak to carry out the equiconsistency of ZF and GB.] 2. I've been somewhat sloppy about the formal details of our length omega constructions. [One occurred in phase B, and one will be about to be presented in phase C.] What's going on is the following: There is some inductive condition I(x,n). There is some next-step condition S(x,y,n). We have the following holding in our model N: 1) There is an x such that I(x,0). 2) For every n in omega, for every x: If I(x,n) there is a y such that S(x,y,n) and I(y, n+1). In that case there is a class of N which is a function f with domain omega such that: 1) For all n, I(f(n),n); 2) For all n, S(f(n), f(n+1), n). Roughly, one takes f(0) to be the L-least x such that I(x,0). One takes f(n+1) to be the L-least y such that S(f(n),y,n) and I(y, n+1). I haven't been too explicit about spelling out I and S precisely, but they are there in the background and would need to be fully spelled out in a completely formal proof. This ends my series of brief remarks and with it letter C7. From solovay@ccrwest.org Thu Mar 15 12:05:37 2001 Return-Path: Received: from diamond.boisestate.edu (IDENT:root@diamond.boisestate.edu [132.178.208.127]) by catseye.idbsu.edu (8.9.3/8.9.3) with ESMTP id MAA00743 for ; Thu, 15 Mar 2001 12:05:36 -0700 Received: from bluenote.ccrwest.org (smap@bluenote.ccrwest.org [192.203.205.129]) by diamond.boisestate.edu (8.9.3/8.9.3) with SMTP id SAA07658 for ; Wed, 14 Mar 2001 18:38:01 -0700 Received: by bluenote.ccrwest.org (4.1/CCRWEST-I1.20) id AA05245; Wed, 14 Mar 2001 17:38:00 PST Received: from ccrwest.ccrwest.org(192.203.205.65) by bluenote.ccrwest.org via smap (V1.3) id sma005235; Wed, 14 Mar 2001 17:37:41 -0800 Received: from makam.ccrwest.org by ccrwest.ccrwest.org (4.1/CCRWEST-2.13) id AA18360; Wed, 14 Mar 2001 17:37:41 PST Received: from localhost by makam.ccrwest.org (4.1/ccrwest-1.6) id AA21977; Wed, 14 Mar 2001 17:37:39 PST Date: Wed, 14 Mar 2001 17:37:39 -0800 (PST) From: "Robert M. Solovay" To: Randall Holmes Subject: That's all folks. Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: R Well the proof survived close scrutiny. Though not without another glitch explained in letter C9. Towards the end I was somewhat neglectful that the maps that move xi_alpha to xi_beta must also move h_{n,alpha} to h_{n,beta}. [This terminology will be explained in letter C9. As usual, comments and questions will be most welcome. --Bob From solovay@ccrwest.org Thu Mar 15 12:05:37 2001 Return-Path: Received: from diamond.boisestate.edu (IDENT:root@diamond.boisestate.edu [132.178.208.127]) by catseye.idbsu.edu (8.9.3/8.9.3) with ESMTP id MAA00746 for ; Thu, 15 Mar 2001 12:05:37 -0700 Received: from bluenote.ccrwest.org (smap@bluenote.ccrwest.org [192.203.205.129]) by diamond.boisestate.edu (8.9.3/8.9.3) with SMTP id SAA07672 for ; Wed, 14 Mar 2001 18:41:01 -0700 Received: by bluenote.ccrwest.org (4.1/CCRWEST-I1.20) id AA05362; Wed, 14 Mar 2001 17:41:00 PST Received: from ccrwest.ccrwest.org(192.203.205.65) by bluenote.ccrwest.org via smap (V1.3) id sma005345; Wed, 14 Mar 2001 17:40:12 -0800 Received: from makam.ccrwest.org by ccrwest.ccrwest.org (4.1/CCRWEST-2.13) id AA18377; Wed, 14 Mar 2001 17:40:12 PST Received: from localhost by makam.ccrwest.org (4.1/ccrwest-1.6) id AA21988; Wed, 14 Mar 2001 17:40:11 PST Date: Wed, 14 Mar 2001 17:40:11 -0800 (PST) From: "Robert M. Solovay" To: Randall Holmes Subject: Letter C11 Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: R We are almost done. Let T_omega be the union of the T_n's. Our construction ensures that T_omega is a complete theory. That is, for any sentence sof L_omega, there is a sentence s' such that: 1) s iff s' is an axiom of T_omega; 2) One of s', \neg s' is an axiom of T_omega. [Let the xi-support of s be {alpha_0, ..., alpha_{n-1}}. Then roughly speaking, s' is obtained from s by replacing xi_{alpha_i} by xi_i.] In the obvious sense, the xi_alphas are fully indiscernible in T_omega. We use T_omega as an EM-blueprint, and construct a model M* generated by indiscernibles xi_i (for i in Z) [Here Z is the full ring of integers (both positive, negative, and zero).] There is an obvious automorphism of M [that moves xi_i to xi_{i+1}]. The actions that we have taken during the course of the construction ensure that the ordinals of M*such that they and their predecessors are fixed by j are just those denoted by \bar{alpha} for some ordinal alpha in N. So M* yields a model of NFU* in the usual way. From solovay@ccrwest.org Thu Mar 15 12:05:37 2001 Return-Path: Received: from diamond.boisestate.edu (IDENT:root@diamond.boisestate.edu [132.178.208.127]) by catseye.idbsu.edu (8.9.3/8.9.3) with ESMTP id MAA00749 for ; Thu, 15 Mar 2001 12:05:37 -0700 Received: from bluenote.ccrwest.org (smap@bluenote.ccrwest.org [192.203.205.129]) by diamond.boisestate.edu (8.9.3/8.9.3) with SMTP id SAA07662 for ; Wed, 14 Mar 2001 18:39:02 -0700 Received: by bluenote.ccrwest.org (4.1/CCRWEST-I1.20) id AA05296; Wed, 14 Mar 2001 17:39:01 PST Received: from ccrwest.ccrwest.org(192.203.205.65) by bluenote.ccrwest.org via smap (V1.3) id sma005255; Wed, 14 Mar 2001 17:38:14 -0800 Received: from makam.ccrwest.org by ccrwest.ccrwest.org (4.1/CCRWEST-2.13) id AA18369; Wed, 14 Mar 2001 17:38:13 PST Received: from localhost by makam.ccrwest.org (4.1/ccrwest-1.6) id AA21980; Wed, 14 Mar 2001 17:38:12 PST Date: Wed, 14 Mar 2001 17:38:12 -0800 (PST) From: "Robert M. Solovay" To: Randall Holmes Subject: Letter C9 Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: R Randall, An irritating little glitch has appeared in the proof. I'm pretty sure I know how to fix it. But this requires many minor changes in the earlier letters. The purpose of this letter is to indicate the changes needed. [The place where the changes are needed is at the point in the argument where one introduces a new h_n function to make the divergent terms not strongly Cantorian.] In place of the function symbol h_n of the prior iteration of the argument, now there will be a constant h_{n,alpha} [one such constant for each n in omega and each ordinal alpha]. This constant should be thought of as h_n(xi_alpha). This will explain the various changes and definitions that follow. The first changes occur in letter C2 [I always refer to the revised versions of letters C1 through C7.] In describing the language L_alpha, I say that there will be one place functions h_n for n < alpha. But now, instead, there will be constants h_{n,gamma} for n < alpha and gamma < kappa. [Later when we shift to working within N, kappa will be replaced by OR--the class of ordinals of N.] When defining the sets which correspond to the various elements of our language, the clause that corresponds to h is: <> = <4,n,alpha>; The notion of h-support is defined inductively as usual. The key clause is that the h-support of h_{n,alpha} is {n}. Similarly, one must define the xi-support of this term. It is {alpha}. In various places, but notably in discussing the thinning condition, we have to consider the following situation. beta_1 and beta_2 are lbfps with beta_1 < beta_2. f is an order preserving map from beta_1 to beta_2. We want to use f to define a map f* from the terms and formulas of L_n \cap V(beta_1) to those of L_n \cap V(beta_2). The definition of f* is inductive and for the most part straightforward. The clauses we want to emphasize are: Let gamma < beta_1. Let i < n. Then f*(xi_gamma) = xi_{f(gamma)}; f*(h_{i,gamma}) = h_{i,f(gamma)}. What takes the place of the prior axioms on h_n are the following: h_{n,alpha} is an ordinal; h_{n,alpha} \leq xi_alpha. I think that does it. This ends letter C9. From solovay@ccrwest.org Thu Mar 15 12:05:37 2001 Return-Path: Received: from diamond.boisestate.edu (IDENT:root@diamond.boisestate.edu [132.178.208.127]) by catseye.idbsu.edu (8.9.3/8.9.3) with ESMTP id MAA00752 for ; Thu, 15 Mar 2001 12:05:37 -0700 Received: from bluenote.ccrwest.org (smap@bluenote.ccrwest.org [192.203.205.129]) by diamond.boisestate.edu (8.9.3/8.9.3) with SMTP id SAA07668 for ; Wed, 14 Mar 2001 18:40:01 -0700 Received: by bluenote.ccrwest.org (4.1/CCRWEST-I1.20) id AA05341; Wed, 14 Mar 2001 17:40:00 PST Received: from ccrwest.ccrwest.org(192.203.205.65) by bluenote.ccrwest.org via smap (V1.3) id sma005333; Wed, 14 Mar 2001 17:39:26 -0800 Received: from makam.ccrwest.org by ccrwest.ccrwest.org (4.1/CCRWEST-2.13) id AA18374; Wed, 14 Mar 2001 17:39:26 PST Received: from localhost by makam.ccrwest.org (4.1/ccrwest-1.6) id AA21985; Wed, 14 Mar 2001 17:39:25 PST Date: Wed, 14 Mar 2001 17:39:24 -0800 (PST) From: "Robert M. Solovay" To: Randall Holmes Subject: Letter C10 Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: R Randall, Before I start discussing the transition from T_n to T_{n+1}, I wanted to record another claim that we have to inductively maintain. As we increase the language, new instances of axiom groups 4 and 5 will arise. They are assumed to be added to the language. So we are given [a Godel number for] the theory T_n, and we have to define T_{n+1}. We first describe a certain set of terms of the language L_n, W_0. W_0 will consist of all closed terms in V(alpha_n) whose xi-support is included in {0, ... , n-2}. [If n <= 1, construe {0, ..., n-2} as the empty set.]. If t \in W_0, let t* be the term obtained by replacing xi_i by xi_{i+1} throughout t [for 0 \leq i < n-1]. [And similarly for the h terms that are subterms of t: i.e, replace h_{j,i} by h_{j,i+1}.] This whole phase is pretty vacuous for n=0. W_0 is then the empty set. T_n will decide the truth value of the sentence t=t*. If it decides that it is true, we say that t is "Cantorian". Let W_1 be the set of Cantorian terms t from W_0 such that T_n believes that "t is an ordinal". We put an equivalence relation == on W_1 as follows: t_1 == t_2 iff T_n decides "t_1 = t_2". Let W_2 be the set of equivalence classes of W_1 under this equivalence relation. We linearly order W_2 as follows: The equivalence class of t_1 is <* the equivalence class of t_2 if T_n decides t_1 < t_2 [using the usual ordering of ordinals]. It is indeed clear that this gives a linear ordering of equivalence classes. In fact, this gives a well-ordering of equivalence classes. We leave the detailed proof of this to the reader. The essential points are (a) Any set of axioms of T_n has a [set] model; (b) The "least ordinal principle" which is part of the axioms of T_n. We say that a t \in W_1 is divergent if the following theory is consistent: (a) The axioms of T_n; (b) For each ordinal gamma, an axiom asserting "t > \bar{gamma}. It is evident that the theory just described is Sigma_2, since T_n is. So working within N, we can tell which terms are divergent. For all we know there could be no divergent terms. But at least the following claims are clear: (a) Whether or not a term t is divergent depends only on the equivalence class of t. (b) If t_1 and t_2 are elements of W_1 and t_1 is divergent, and the equivalence class of t_1 is <* the equivalence class of t_2, then t_2 is divergent. If there are no divergent terms, we set T_{n,1} = T_n. If there are divergent terms, we pick an equivalence class of such divergent terms which is <* minimal. We then pick a t in this equivalence class [using the canonical well-ordering that exists in the model N of V=L]. We then get T_{n,1} from T_n by adjoining the following list of axioms: If gamma is an ordinal, we have the axiom "t > \bar{gamma}". Since t is divergent, the theory T_{n,1} is consistent. We need to check that T_{n,1} has the thinning property. When I introduced T_0, I should have assigned the reader the following easy exercise: T_0 has the thinning property. The proof is not difficult. The only axiom groups that need any thought are axiom groups 3, 4 and 5. We shall suppose that [as part of our inductive hypothesis] we know that T_n has the thinning property. We also will have an indiscernibility property that says [roughly] that all increasing n-tuples of xi's look alike. Using this, it is easy to see that T_{n,1} has the thinning property. We next take steps to insure that the term t [our minimal divergent term] is not strongly Cantorian. In doing this we will define a new theory T_{n,2} in the language L_{n+1}. The axioms of T_{n+1} consist of the following: (a) The axioms of T_{n,1}. (b) The new axioms of axiom group 4 [cf. letter C6] that apply to terms of the form h_{n,alpha}. [Cf. letter C9 for the revised form of these.] (c) The new instances of Axiom group 5 ["the least ordinal principle"] that occur since we have enlarged the language. (d) For each pair of ordinals alpha and beta with alpha < beta, an axiom that asserts that h_{n,_alpha} < h_{n,beta}. (e) Axioms of the following group will only be added if there is a divergent term in W_1 [so that t is defined]. Let delta_0 < delta_1 < ... < delta_{n-1} be ordinals. Let t[\vec{delta}] be the closed term obtained from t by replacing xi_i by xi_{delta_i} throughout [and similarly for any subterm of t which is an h term.]. Then there will be an axiom h_{n,delta_0}) < t[\vec{delta}]. Claim: T_{n,2} is consistent. Proof: Fix a lbfp beta > alpha_n. It suffices to show that T_{n,2} \cap V(beta) has a model M* [for any such beta]. Let M be a model of T_{n,1} \cap V(beta). We are going to enrich M so as to make it a model of T_{n,2} \cap V(beta). The first thing is to give the meaning of the terns h_{n,delta} (for delta < beta). It will denote the same element of M that \bar{delta} does. Many new closed terms of L_{n+1} will be elements of V(beta). We have to assign a meaning to each such closed term t. We will do this by induction on the rank of t. What we will do is assign a closed term of L_n \cap V(beta) for it to shadow. At the same time, we will be associating to each formula of L_{n+1} a formula of L_n. It is our intention that for the model M*, the formula of L_{n+1} and its shadow in L_n will have the same meaning. Let \sh{s} be the shadow of s. Here is the inductive definition: \sh{\bar{alpha} = \bar{alpha}; \sh{v_alpha} = v_alpha; \sh{xi_alpha} = {xi_alpha}; If i < n, \sh{h_{i,alpha}} = h_{i,alpha}; \sh{h_{n,alpha}} = \bar{alpha}; \sh{ \iota x psi(x)} = \iota x \sh{psi(x)}; \sh{ \neg psi} = \neg \sh{psi}; \sh{t_1 = t_2} = \sh{t_1} = \sh{t_2}; \sh{t_1 \in t_2} = \sh{t_1} \in \sh{t_2; \sh{\exists x psi} = \exists x \sh{psi}; Finally we have to handle infinitary disjunctions. \sh{ \bigvee S} = \bigvee S* where S* = {\sh{s} | s \in S}. This completes our description of M*. Note that for every closed term of M* there is a closed term of M with the dame denotation. Note that by the key new axiom of T_{n,1} the value of t is greater than the values of the h_{n,alpha}'s. It remains to see that M* is a model of T_{n,2} \cap V(beta). But this is straightforward to check. [Our first comment is helpful in verifying the axioms of group (c) and our second in verifying the axioms of group (e). It is easy to check that T_{n,2} has the thinning property. Our next term is to define alpha_{n+1}. Let t be a term in W_1 that is not divergent. Let T_n[t] be the theory in the language L_n obtained from T_n by adding the axioms t > \bar{alpha} for each alpha in OR. By assumption, T_n[t] is inconsistent. Using Sigma_2-replacement, it is easy to see: There is a single ordinal delta_n which is a lbfp such that for any term t in W_1 which is not divergent, T_n[t] \cap V(delta_n) is inconsistent. We take alpha_{n+1} to be the least lbfp which is greater than alpha_n, delta_n, and gamma_n. [The gamma_n's are our standard sequence that cofinalizes OR in N.] Our next theory will make sure that all the n+1-tuples of xi's look alike. T_{n,3} is obtained from T_{n,2} by adding the following axioms: For each sentence s whose xi-support is included in {0,...,n} and each order preserving map f:{0,...,n} --> OR, there will be an axiom A(s,f) defined as follows: Let f* be the map on terms and formulas induced by f. [Recall that f* acts on h terms in a non-trivial fashion.] Then A(s,f) is the sentence s iff f*(s). Claim: T_{n,3} is consistent. Let beta be a lbfp greater than alpha_{n+1}. It's enough to construct a model of T_{n,3} \cap V(beta). Let gamma be a lbfp greater than beta, and let M be a model of T_{n,2} \cap V(gamma). By Erdos-Rado, we can find a subset Y of gamma of order-type beta such that for any s in V(beta) with xi-support included in {0, ..., n} and any order preserving map f: {0, ..., n} --> Y, the truth value of f*(s) has the same value. Let g: beta --> Y be the obvious order isomorphism. We define a model M* by giving the value of the closed term t in L_{n+1} \cap V(beta) in M* to be the value of g*(t) in M. Since T_{n,2} has the thinning property, M* is a model of T_{n,2}. But by choice of Y, it is evident that M* satisfies the additional axioms we added to T_{n,2} to make T_{n,3}. {At least those instances lying in V(beta). The proof of the claim is complete. Claim: T_{n,3} has the thinning property. Proof: Exercise. Left to the reader. We finally are in a position to define T_{n+1}. First let's introduce a set of sentences S_{n+1}: S_{n+1} consists of all sentences of L_{n+1} which lie in V(alpha_{n+1}) and have xi-support included in {0, ..., n}. To each f: S_{n+1} --> {0,1} we introduce a sentence A(f) as follows: A(f) is the infinite conjunction \bigand {s^{f(s) | s \in S_{n+1}} Here if s is a sentence s^0 is \neg s, and s^1 is s. Claim: For some f, T_{n,3) + A(f) is consistent. Proof: Take a branch through T(T_{n,3}) and use the f it defines. Let f be least such that the claim is true. T_{n+1} = T_n together with the axioms s^{f(s)} for s \in S_{n+1}. Claim: T_{n+1} has the thinning property. Proof: Easy and left to the reader. The axioms we added to make T_{n,3} play a crucial role here. This completes our length omega construction, and with it letter C10.