From solovay@math.berkeley.edu Sat Apr 15 16:00 MDT 1995
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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
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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
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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
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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
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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
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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
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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
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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
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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
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From: solovay@math.berkeley.edu (Robert M. Solovay)
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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
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From: solovay@math.berkeley.edu (Robert M. Solovay)
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Date: Wed, 19 Apr 1995 13:11:46 -0700
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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
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Date: Wed, 19 Apr 1995 14:17:04 -0600
From: Randall Holmes
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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
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From: Randall Holmes
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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
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From: solovay@math.berkeley.edu (Robert M. Solovay)
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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
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From: solovay@math.berkeley.edu (Robert M. Solovay)
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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
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From: Randall Holmes
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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
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From: Randall Holmes
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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
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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
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From: Randall Holmes
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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
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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
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From: Randall Holmes
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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
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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
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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
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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
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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
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Date: Fri, 21 Apr 1995 09:08:40 -0600
From: Randall Holmes
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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
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Date: Fri, 21 Apr 1995 09:10:54 -0600
From: Randall Holmes
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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
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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
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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
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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
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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
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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
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From: Randall Holmes
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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
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From: Randall Holmes
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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
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From: Randall Holmes
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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
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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
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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
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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
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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
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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
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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
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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
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From: Randall Holmes
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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
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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
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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
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From: Randall Holmes
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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
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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
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From: Randall Holmes
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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
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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
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From: solovay@math.berkeley.edu (Robert M. Solovay)
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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.
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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
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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
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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
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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
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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
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From: Akihiro Kanamori
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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
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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
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From: Akihiro Kanamori
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Date: Fri, 23 Feb 1996 16:49:12 -0500 (EST)
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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
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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
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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
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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
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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
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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
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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
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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
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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
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From: Randall Holmes
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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
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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
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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
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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
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From: Randall Holmes
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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
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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
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From solovay@math.berkeley.edu Fri Jun 13 19:45 MDT 1997
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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
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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
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From: Randall Holmes
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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
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From: Randall Holmes
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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
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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
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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
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From: Randall Holmes
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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
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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
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From: Randall Holmes
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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
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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
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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
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From: Randall Holmes
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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
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From: Solovay Robert
To: Randall Holmes
Subject: Preliminary announcement
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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
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From: Randall Holmes
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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
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From: Solovay Robert
To: Randall Holmes
Subject: Reply to your letter of July 9th
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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
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From: Solovay Robert
To: Randall Holmes
Subject: Reply to your letter of July 9th
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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
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From: Solovay Robert
To: Randall Holmes
Subject: Re: Reply to your letter of July 9th
In-Reply-To: <9707091924.AA06750@isiax1.isi.it>
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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
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From: Solovay Robert
To: Randall Holmes
Subject: The proof has survived
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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
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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}
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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
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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
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From: "Robert M. Solovay"
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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
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From: "Robert M. Solovay"
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Date: Sat, 22 Apr 2000 19:25:39 -0700 (PDT)
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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
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From: "Robert M. Solovay"
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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
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Date: Mon, 24 Apr 2000 07:50:13 -0600
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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
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From: "Robert M. Solovay"
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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
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From: "Robert M. Solovay"
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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
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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
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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
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---559023410-851401618-957503405=:15840--
From solovay@math.berkeley.edu Sat May 13 22:17:04 2000
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Date: Sat, 13 May 2000 21:16:25 -0700 (PDT)
From: "Robert M. Solovay"
X-Sender: solovay@blue1
To: Randall Holmes
Subject: Letter 6
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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
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---559023410-851401618-958277785=:13792--
From solovay@math.berkeley.edu Mon May 15 23:24:16 2000
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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
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---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
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---559023410-851401618-958454614=:5342--
From solovay@math.berkeley.edu Tue May 16 15:04:43 2000
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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
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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
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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
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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>
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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
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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>
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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
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Date: Mon, 1 Jan 2001 18:35:48 -0800 (PST)
From: "Robert M. Solovay"
X-Sender: solovay@blue2
To: Randall Holmes
Subject: Apologies
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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
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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
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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
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From solovay@ccrwest.org Mon Feb 26 12:40:05 2001
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From: "Robert M. Solovay"
To: "M. Randall Holmes"
Subject: Re: Remarks on your retraction
In-Reply-To: <200102261607.JAA05985@catseye.idbsu.edu>
Message-Id: *