# Set Theory Seminar Archives

This is an archive of some of the abstracts of the Set Theory Seminar of the set theory group.

2016–2017

Organizer: Samuel Coskey

Tuesday, April 4 from 3 to 4pm
Room: MB 124
Speaker: Marion Scheepers (BSU)
Title: Playing an infinitely long game when you have limited memory (IV)

Abstract: We consider a class of infinite games in which player TWO has a winning strategy (based on perfect memory). In prior talks in this series we considered the effect of a limited memory where TWO remembers only the most recent move of ONE and of TWO, or TWO remembers a limited number of prior moves of ONE only.

As in these prior talks we consider the game where ONE chooses a first category subset of a space, and TWO chooses a nowhere dense set each inning. ONE’s sets are strictly increasing from inning to inning. For a fixed k, TWO remembers only the most recent k moves of ONE. We discussed why for k=1 only in the simplest of circumstances TWO has a winning 1-tactic. We also outline how in certain examples TWO had a winning 2-tactic ($k=2$). In this talk we will focus on the case when TWO does not have a winning 2-tactic, but does have a winning k-tactic for some $k>2$.

Tuesday, February 14 from 3 to 4pm
Room: MB 124
Speaker: Sam Dworetzky (BSU)
Title: The classification problem for models of Peano Arithmetic

Abstract: It is well known that the countable models of Peano Arithmetic (PA) are complicated to classify. In this talk we will make this precise using the language of Borel complexity theory. We will use a construction of Gaifman to show there is a Borel reduction from the isomorphism relation on linear orders to the isomorphism relation on models of PA.

Tuesday, January 31 from 3 to 4pm
Room: MB 124
Speaker: Stephanie Potter (BSU)
Title: Classification of vertex-transitive structures

Abstract: In this talk we will consider the Borel complexity of the isomorphism relation on two natural classes of objects: countable, connected, vertex-transitive graphs and countable, vertex-transitive partial orders. We will first discover that the isomorphism relation on vertex-transitive graphs has maximal complexity or, in other words, is Borel complete. This result will then allow us to show that the isomorphism relation on countable, vertex-transitive partial orders is also Borel complete.

Tuesday, January 17 from 3 to 4pm
Room: MB 124
Speaker: Marion Scheepers (BSU)
Title: Playing an infinitely long game when you have limited memory (III)

Abstract: We consider a class of infinite games in which player TWO has a winning strategy (based on perfect memory). In prior talks in this series we considered the effect of a limited memory where TWO remembers only the most recent move of ONE and of TWO. We now discuss the effect of TWO for a fixed positive integer k remembering only the most recent k or fewer moves of ONE.

Tuesday, November 29 from 3 to 4pm
Room: MB 124
Speaker: John Clemens (BSU)
Title: Kleene’s recursion theorem and perfect kernels

Abstract: I will continue discussing applications of Kleene’s Second Recursion Theorem in Descriptive Set Theory, and how it can be used with other structural properties of point classes to produce definable fixed points of set operations. In particular I will prove Kreisel’s theorem on perfect kernels. Recall that any closed set may be decomposed into the union of a countable set and a perfect set; the latter is called the perfect kernel of the set. Kernel’s theorem gives sharp complexity bounds on the perfect kernel and can be used to show that the map sending a code for a closed set to a code for its perfect kernel is suitably definable.

Tuesday, November 8 from 3 to 4pm
Room: MB 124
Speaker: Randall Holmes (BSU)
Title: Constructing proofs with a dependent type checker, II

Abstract: The talk will be a continuation of the previous one.

Tuesday, October 25 from 3 to 4pm
Room: MB 124
Speaker: Randall Holmes (BSU)
Title: Constructing proofs with a dependent type checker

Abstract: I’ll discuss my latest theorem proving work, and present examples. I’ll explain what dependent type theory is and how a type checker for a dependent type theory can be a theorem prover. The talk should be accessible to students.

Tuesday, October 11 from 3 to 4pm
Room: MB 124
Speaker: John Clemens (BSU)
Title: The Recursion Theorem in Set Theory, II

Abstract: We continue the lecture from the previous session.

Tuesday, September 27 from 3 to 4pm
Room: MB 124
Speaker: John Clemens (BSU)
Title: The Recursion Theorem in Set Theory

Abstract: Kleene’s Recursion Theorem is a simple yet powerful result about computable functions, which asserts the existence of functions which “know their own code” in some suitably nice enumeration of the computable functions. It can be used to find fixed points for operations on computable functions. Less well-known is that the Recursion Theorem can be applied to other Polish spaces (instead of the integers) to produce fixed points for set operations. I will give a brief introduction to the theory of computable functions and the original Recursion Theorem, and then discuss the basics of effective descriptive set theory necessary to extend it to more general Polish spaces. As time permits I will sketch some of the consequences in descriptive set theory, such as finding least fixed points of monotone set operations and Moschovakis’s Coding Lemma.

Tuesday, September 13 from 3 to 4pm
Room: MB 124
Speaker: Samuel Coskey (BSU)
Title: Classifying automorphisms of countable trees

Abstract: We summarize some of the results from Kyle Beserra’s master’s thesis. In Serre’s study of trees and their automorphisms, he observed that the automorphisms all lie in one of three classes: invert an edge, shift a bi-infinite path, or fix a subtree pointwise. But of course there are many types of automorphisms within each of these classes. So it is natural to ask just how complex is the classification of tree automorphisms? And what is the complexity of each of Serre’s three classes? We can make these questions formal using the language Borel complexity theory. In this talk we answer the question for regular trees.

2015–2016

Organizer: Samuel Coskey

Wednesday, April 20 from 3 to 4pm
Room: MB 124
Speaker: Marion Scheepers (BSU)
Title: Ramsey theory and the Borel Conjecture

Abstract: The Borel Conjecture states that certain measure zero sets of real numbers are countable. This statement can be converted to a statement that a set of real numbers is one of these special measure zero sets if, and only if, a special associated structure satisfies a version of Ramsey’s Theorem. We discuss this connection, and explore a range of statements equivalent to the alluded to version of Ramsey’s Theorem.

Wednesday, April 6 from 3 to 4pm
Room: MB 124
Speaker: Samuel Coskey (BSU)
Title: López-Escobar’s theorem

Abstract: We present a now classical theorem of López-Escobar which connects descriptive set theory and countable model theory. We then briefly introduce the theory of metric structures, and give a generalization of the theorem to this context. This is joint work with Martino Lupini.

Wednesday, March 2 from 3 to 4pm
Room: MB 124
Title: Jónsson successors of singular cardinals

Abstract: We say that a cardinal $\lambda$ is a Jónsson cardinal if it satisfies the following weak Ramsey-type property: given any coloring $F:[\lambda]^{<\omega}\to \lambda$ of the finite subsets of $\lambda$ in $\lambda$-many colors, there exists a set $H\in[\lambda]^\lambda$ such that the range of $F\upharpoonright [H]^{<\omega}$ is a proper subset of $\lambda$. One of the big open questions in combinatorial set theory is whether or not the existence of a singular cardinal $\mu$ such that $\mu^+$ is a Jónsson cardinal is even consistent. The goal of this talk is to explain why this problem has proven so difficult, and to (time permitting) briefly survey ongoing research in the area.

Wednesday, February 17 from 3 to 4pm
Room: MB 124
Speaker: John Clemens (BSU)
Title: Countable sets of reals and primeness

Abstract: Abstract: I will sketch the proof of the result of Kanovei-Sabok-Zapletal that the equivalence relation of equality of countable sets of reals (=+) is prime, i.e., for any other Borel equivalent relation F, either =+ is Borel-reducible to F or any homomorphism from =+ to F collapse on a set as complicated as =+ itself. I will also discuss limitations in extending the techniques to other isomorphism problems for classes of countable structures.

Wednesday, February 3 from 3 to 4pm
Room: MB 124
Speaker: Randall Holmes (BSU)
Title: Consistency of new foundations III

Abstract: The third talk in the series described below.

Wednesday, January 20 from 3 to 4pm
Room: MB 124
Speaker: Randall Holmes (BSU)
Title: Consistency of new foundations II

Abstract: This will be the second of three or four talks which will give an account of my consistency proof for Quine’s system of set theory New Foundations (proposed by W. v.O. Quine in 1937). The first talk covered preliminaries about the typed theory of sets (an unproblematic variant of the usual set theory), the modification of the definition of the typed theory of sets to give New Foundations, and the relevant known results about the model theory of NF and the relative consistency of related systems. The construction is via a Fraenkel-Mostowski construction of a model of ZFA (the usual set theory with atoms) without Choice, so this talk will be a discussion of this kind of construction, with an example relevant to the eventual main construction. All work done in the talk will actually be done in the usual set theory or variants (give or take the existence of atoms or the assumption of the axiom of choice).

Wednesday, December 9 from 3 to 4pm
Room: MP 207
Speaker: Randall Holmes (BSU)
Title: Consistency of New Foundations

Abstract: This will be the first of three or four talks which will give an account of my consistency proof for Quine’s system of set theory New Foundations (proposed by W. v.O. Quine in 1937). The first talk will cover preliminaries about the typed theory of sets (an unproblematic variant of the usual set theory), the modification of the definition of the typed theory of sets to give New Foundations, and the relevant known results about the model theory of NF and the relative consistency of related systems. The construction is via a Fraenkel-Mostowski construction of a model of ZFA (the usual set theory with atoms) without Choice, so the next point will be a discussion of this kind of construction, with an example relevant to the eventual main construction. All work done in the talk will actually be done in the usual set theory or variants (give or take the existence of atoms or the assumption of the axiom of choice).

Wednesday, November 11 from 3 to 4pm
Room: MP 207
Speaker: John Clemens (BSU)
Title: Relative primeness of equivalence relations

Abstract: Let $E$ and $F$ be equivalence relations on the spaces $X$ and $Y$. We say that $E$ is prime to $F$ if: whenever $\varphi: X \rightarrow Y$ is a homomorphism from $E$ to $F$, there is a continuous embedding $\rho$ from $E$ to itself so that the range of $\varphi \circ \rho$ is contained in a single $F$ class. That is to say, $\varphi$ is constant (up to $F$-equivalence) on a set on which $E$ maintains its full complexity with respect to Borel reducibility. When $E$ is prime to $F$, $E$ fails to be Borel-reducible to $F$ in a very strong way. I will discuss this notion and show that many non-reducibility results in the theory of Borel equivalence relations can be strengthened to produce primeness results. I will also discuss the possibility of new types of dichotomies involving the notion of primeness.

Wednesday, October 30 from 3 to 4pm
Room: MP 207
Speaker: Marion Scheepers (BSU)
Title: Playing an infinitely long game when you have limited memory, II

Abstract: The talk from last time concludes.

Wednesday, October 14 from 3 to 4pm
Room: MP 207
Speaker: Marion Scheepers (BSU)
Title: Playing an infinitely long game when you have limited memory

Abstract: In some infinite games one of the two players has a clear advantage, provided that the player has perfect memory. In this talk we briefly survey some results when the winning perfect information player has a limited memory.

Wednesday, September 30 from 3 to 4pm
Room: MP 207
Speaker: Liljana Babinkostova (BSU)
Title: The selective strong screenability game

Abstract: Selective versions of screenability and of strong screenability coincide in a large class of spaces. We show that the corresponding games are not equivalent in even such standard metric spaces as the closed unit interval. We identify sufficient conditions for ONE to have a winning strategy, and necessary conditions for TWO to have a winning strategy in the selective strong screenability game.

Wednesday, September 16 from 3 to 4pm
Room: MP 207
Speaker: John Clemens (BSU)
Title: Isomorphism of homogeneous structures

Abstract: The theory of Borel reducibility of equivalence relations can be used to gauge the complexity of the isomorphism problem for a collection of countable structures. Certain classes, such as that of graphs and trees, are known to have an isomorphism problem of maximal complexity. We may also consider only the homogeneous structures, those whose automorphism group acts transitively on the structure. I will discuss the question of when the isomorphism problem for a collection of homogeneous structures is as complicated as that for all such structures. This may be viewed as asking when information may be coded into a structure without using “local” coding. In particular, we can show that this is true for graphs, but not for trees.

Wednesday, September 2 from 3 to 4pm
Room: MP 207
Speaker: Samuel Coskey (BSU)
Title: Amalgamation properties and conjugacy

Abstract: In this talk we aim to classify automorphisms of ultrahomogeneous structures up to conjugacy. (Ultrahomogeneous structures are highly self-symmetric and can be viewed as a kind of generic limit of their finite pieces.) In past talks I gave several examples of homogeneous structures and settled the classification problem for their automorphisms. After reviewing the classical combinatorial framework, we will generalize the methods that worked for specific examples to broader classes of ultrahomogeneous structures.

2014–2015

Organizer: Samuel Coskey

Wednesday, April 29 from 3 to 4pm
Room: Math 124
Speaker: Stuart Nygard (BSU)
Title: The density topology

Abstract: In the Euclidean topology, open sets are defined by unions of open intervals. Can we remove some “small” set from the intervals and still have a meaningful topology? Yes. We define a topology using sets that have locally full measure. That is, a set will have an open neighborhood around a point if “almost every” point nearby belongs to the set. We will show how the topology naturally arises on R and other spaces. No knowledge of Lebesgue measure or topology is assumed.

Wednesday, April 8 from 3 to 4pm
Room: Math 124
Speaker: Samuel Coskey (BSU)
Title: López-Escobar’s theorem for metric structures

Abstract: The classical López-Escobar theorem states that any Borel class of countable structures may be axiomatized using an appropriate infinitary logic. One application of this theorem is to relate topological and model-theoretic versions of Vaught’s conjecture. In this talk we present a variant of López-Escobar’s theorem for metric structures, which implies that Borel classes of separable metric structures may be axiomatized in the appropriate infinitary continuous logic. As a consequence we obtain a new implication between the topological Vaught conjecture and a version for metric structures. This is joint work with Martino Lupini.

Wednesday, April 1 from 3 to 4pm
Room: Math 124
Speaker: Randall Holmes (BSU)
Title: Preliminaries for Proving the Consistency of NF, IV

Abstract: We will continue our previous discussion of the machinery of a consistency proof of Quine’s set theory New Foundations.

Wednesday, March 18 from 3 to 4pm
Room: Math 124
Speaker: Randall Holmes (BSU)
Title: Preliminaries for Proving the Consistency of NF, III

Abstract: We will continue our previous discussion of the machinery of a consistency proof of Quine’s set theory New Foundations.

Wednesday, March 4 from 3 to 4pm
Room: Math 124
Speaker: Samuel Coskey (BSU)
Title: An introduction to continuous logic

Abstract: Continuous logic is a proper generalization of first order logic where the usual binary truth values are replaced by the unit interval $[0,1]$. The models for this logic are metric structures, which are metric spaces together with continuous functions and $[0,1]$-valued relations. Just as ordinary logic has typical applications in discrete math, continuous logic has applications in analysis. In this talk we will introduce just the basic concepts and theory of continuous logic.

Wednesday, February 18 from 3 to 4pm
Room: Math 124
Speaker: Randall Holmes (BSU)
Title: Preliminaries for Proving the Consistency of NF, II

Abstract: We will continue our previous discussion of the machinery of a consistency proof of Quine’s set theory New Foundations. Once again, no particular familiarity with New Foundations is presupposed.

Wednesday, February 11 from 3 to 4pm
Room: Math 124
Speaker: Randall Holmes (BSU)
Title: Preliminaries for Proving the Consistency of NF

Abstract: We will discuss some preliminary machinery intended for use in a consistency proof of Quine’s set theory New Foundations.
No particular familiarity with New Foundations is presupposed.

Wednesday, February 4 from 3 to 4pm
Room: Math 124
Speaker: Samuel Coskey (BSU)
Title: Conjugacy and homogeneous graphs, II

Abstract: This talk is continued from last semester. We study the conjugacy relation on the automorphism group of a homogeneous structure from the point of view of Borel complexity theory. In particular, we present three examples of homogeneous graphs whose corresponding conjugacy problems have three different complexities.

Wednesday, January 28 from 3 to 4pm
Room: Math 124
Speaker: Andrés Caicedo (BSU)
Title: Some topological partition properties of $\omega_1$, part II

Abstract: We discuss some new results on the topological partition calculus of ordinals less than or equal to $\omega_1$. This is joint work with Jacob Hilton.

Wednesday, January 21 from 3 to 4pm
Room: Math 124
Speaker: Andrés Caicedo (BSU)
Title: Some topological partition properties of $\omega_1$

Abstract: We present some classical and new positive results on the topological version of partition relations involving $\omega_1$. We also use the topic as an excuse to review some combinatorics of stationary sets, including Fodor’s lemma.

Wednesday, December 10 from 3 to 4pm
Room: Math 226
Speaker: Andrés Caicedo (BSU)
Title: Co-analytic uniformization

Abstract: It is an easy consequence of the axiom of choice that if X is an arbitrary set and R is a binary relation on X (a subset of $X^2$) then R admits a uniformization, that is, there is a function f whose domain is $\{x \in X : \text{there is a } y \in X \text{ with } x R y\}$ and such that for all x in its domain, x R f(x).

If X is the set of reals, and R is a reasonably definable relation, one might expect that the existence of such a function f can actually be established without using the axiom of choice.

We sketch a classical result independently due to Novikov and Kondo showing that this is indeed the case if R is Borel (and even if it is “slightly” more complicated than Borel).

Wednesday, October 1 from 3 to 4pm
Room: Math 226
Speaker: Andrés Caicedo (BSU)
Title: Ramsey theory of very small countable ordinals II

Abstract: We examine a closed version of the pigeonhole principle for ordinals, and use it to draw upper bounds on closed Ramsey numbers.

Wednesday, September 24 from 3 to 4pm
Room: Math 226
Speaker: Andrés Caicedo (BSU)
Title: Ramsey theory of very small countable ordinals

Abstract: We present a brief introduction to classical Ramsey theory, and discuss two extensions in the context of ordinals. We limit ourselves to small countable ordinals, emphasizing those smaller than $\omega^2$.

Wednesday, September 10 from 3 to 4pm
Room: Math 226
Speaker: Thomas Forster (Cambridge)
Title: WQOs and BQOs – an Introductory Talk

Abstract: A WQO is a transitive reflexive relation with no infinite antichains and no infinite strictly descending chains. In this introductory talk (very few proofs!) for a general mathematical audience i shall try to show some of the many places that WQOs have spread their tentacles into, how they give rise to BQOs, the connections with finite combinatorics (Seymour-Robertson theorem), undecidability results in arithmetic and other fun stuff. I’ll even tell you what the two TLAs stand for.

Wednesday, September 3 from 3 to 4pm
Room: Math 226
Speaker: Samuel Coskey (BSU)
Title: Conjugacy and homogeneous graphs

Abstract: Before studying the conjugacy problem in a given Polish group G, it is natural to ask what is the complexity of the conjugacy equivalence relation. We study this relation in the case when G is the automorphism group of a homogeneous graph (directed or undirected). The homogeneous graphs have been classified by Cherlin and we will briefly run through the complete list. Then we will give examples where the complexity of the conjugacy relation on G is smooth, complete, and in between.

2013–2014

Organizer: Samuel Coskey

Tuesday, April 22 from 2 to 3pm
(This talk is joint with the math department Colloquium. Refreshments will be served in MB 226 beforehand.)
Room: ILC 204
Speaker: Martino Lupini (York University)
Title: An invitation to sofic groups

Abstract: The class of countable discrete groups known as sofic groups has drawn in the last ten years the attention of an increasing number of mathematicians in different areas of mathematics. Many long-standing conjectures about countable discrete groups have been settled for sofic groups. Despite the amount of research on this subject, several fundamental questions remain open, such as: Is there any group which is not sofic? In my talk I will give an overview of the theory of sofic groups and its applications.

Tuesday, April 15 from 2 to 3pm
Room: Mathematics 136
Speaker: Rodrigo Roque Dias (BSU and São Paulo)
Title: Supertightness and point-picking games

Tuesday, March 18 from 2 to 3pm
Room: Mathematics 136
Speaker: Dillon Wardwell (BSU)
Title: Zariski Selection Principles

Abstract: Let R be a commutative ring with unity, and Spec(R) denote the collection of prime ideals of R, also called the prime spectrum of R. The prime spectrum becomes a topological space when endowed with the Zariski topology. We explore what effect the algebraic properties of the underlying ring R have on which classical selection principles hold in the spectrum via winning strategies in related infinite games.

Tuesday, February 18 from 2 to 3pm
Room: Mathematics 136
Speaker: Andrés Caicedo (BSU)
Title: An absoluteness result

Abstract: We present examples of some (Ramsey-theoretic) theorems of ZFC (whose standard proofs make blatant use of choice) that can be established in ZF as well. We provide a combinatorial proof of one of these statements, and also a general metamathematical argument (due to Asaf Karagila) showing they must be provable in ZF based simply on their logical complexity.

Tuesday, February 11 from 2 to 3pm
Room: Mathematics 136
Speaker: Samuel Coskey (BSU)
Title: A longer Choquet game

Abstract: A topological space is Polish if it is separable and completely metrizable. Since many properties common to the Cantor space $2^\omega$ and the Baire space $\omega^\omega$ generalize to Polish spaces, they are naturally the subject of study in descriptive set theory. Among separable and normal spaces, Choquet characterized the Polish ones as those where the second player has a winning strategy in the so-called Choquet game. In this talk, we will study the generalization of this property to spaces of higher weight, where the Choquet game is replaced by an analogous game of longer length. This is joint work with Philipp Schlicht.

Tuesday, February 4 from 2 to 3pm
Room: Mathematics 136
Speaker: Rodrigo Roque Dias (BSU and São Paulo)
Title: The point-open game in products

Abstract: The point-open game was introduced independently by R. Telgársky (1975) and F. Galvin (1978). In Telgársky’s paper, he asks whether the property “player One has a winning strategy in the point-open game” is finitely productive. In this talk we will answer Telgársky’s question in the affirmative and discuss variations of this result in the context of Rothberger spaces; in particular, we will show that a space in which player One has a winning strategy in the point-open game is productively Rothberger. If time permits, we will also discuss how the proofs of the above results can be adapted to the context of Menger spaces and the Menger game.

Tuesday, January 28 from 2 to 3pm
Room: Mathematics 136
Speaker: Randall Holmes (BSU)
Title: On hereditarily small sets

Abstract: Jech showed in 1982 that $H(\omega_1)$, the collection of hereditarily countable sets, is a set in ZF. In the absence of choice, the existence of $H(\omega_1)$ is not obvious; but he showed that the rank of any hereditarily countable set is less than $\omega_2$. Forster has remarked that Jech’s proof generalizes to show the existence of $H(\kappa)$ when $\kappa$ is an $\aleph$ cardinal (a cardinal of well-orderable sets). We have shown the existence of $H(\kappa)$, the collection of all sets hereditarily of size less than $\kappa$, in ZF without Choice, for any cardinal $\kappa$ at all, by a not altogether trivial generalization of Jech’s technique.

Monday, December 9 from 3 to 4pm
Room: Mathematics 136
Speaker: Samuel Coskey (BSU)
Title: Countable abelian groups and conjugacy relations

Abstract: How hard is it to classify the countable abelian groups? While it is known that the classification of arbitrary countable groups is as complex as that for arbitrary countable structures, the analogous question for countable abelian groups remains open. I will present a partial result of Hjorth which says that the classification of countable abelian groups is fairly complex in that it is not Borel. I will also show how this result is related to a question that arises in the study of conjugacy relations.

Monday, November 18 from 3 to 4pm
Room: Mathematics 136
Speaker: Bruno Pansera (University of Messina, Italy)
Title: Dense families and selection principles

Abstract: We present a classiﬁcation for the study of classic and new properties involving diagonalizations of dense families in topological spaces.

Monday, November 4 from 3 to 4pm
Room: Mathematics 136
Speaker: Bruno Pansera (University of Messina, Italy)
Title: On the Urysohn number of a topological space

Abstract: We present some variations of Arhangel’skii inequality. This is joint work with Maddalena Bonanzinga.

Monday, October 28 from 3 to 4pm
Room: Mathematics 136
Title: Inner Model Theory II: Baby Mice

Abstract: In this talk, we develop the relationship between $L$ and $\mathcal{M}_0^\sharp$.

Monday, October 21 from 3 to 4pm
Room: Mathematics 136
Title: Inner Model Theory I: Setting and Tools

Abstract: This talk is the first in a sequence of talks, the purpose of which is to outline some of the motivations, tools, and themes of inner model theory. As the purpose is expository, and in order to make this a manageable project, the setting will be non-fine structural. Special attention will be paid to $\mathcal{M}_0^{\sharp}$ and $L$.

In this particular talk, our goal is to set the stage and discuss some of the basic tools of inner model theory. Here, the focus will be on elementary embeddings, the ultraproduct construction, linear iterations, and making concrete the idea of canonical inner models for “small” large cardinals.

Monday, October 14 from 3 to 4pm
Room: Mathematics 136
Speaker: Rodrigo R. Dias (Universidade de São Paulo and BSU)
Title: Alster spaces and the Menger game

Abstract: We answer a question of Frank Tall by showing that the existence of a winning strategy for player Two in the Menger game is strictly stronger than the Alster property. This is a joint work with Leandro F. Aurichi.

Monday, October 7 from 3 to 4pm
Room: Mathematics 136
Speaker: Andrés Caicedo (BSU)
Title: Second incompleteness

Abstract: We sketch an essentially model theoretic proof (due to Woodin) of the second incompleteness theorem for ZF.

Monday, September 30 from 3 to 4pm
Room: Mathematics 136
Speaker: Samuel Coskey (BSU)
Title: Splitting relations

Abstract: If $A$ and $B$ are infinite subsets of $\mathbb N$, we say that $A$ splits $B$ if both $A\cap B$ and $A^c\cap B$ are infinite. We call a family $\mathcal F$ of infinite subsets of $\mathbb N$ a splitting family if for every infinite set $B$ there is $A\in\mathcal F$ such that $A$ splits $B$.

In this talk, we consider some natural generalizations of splitting families, namely, $\mathcal F$ is said to be an $n$-splitting family if for every sequence of infinite sets $B_1,\ldots,B_n$ there exists $A\in\mathcal F$ which splits them all. Although the least cardinality of an $n$-splitting family is the same size for all $n$, we will show that they are in fact distinct notions.

Specifically, we will show that the $n$-splitting relations form a chain in the Borel Tukey ordering on relations of this type. We will also show how to use similar examples to find an infinite antichain in the Borel Tukey ordering.

Presenting joint work with Juris Steprāns.

Monday, September 16 from 3 to 4pm
Room: Mathematics 136
Speaker: Philipp Kleppmann (Cambridge)
Title: Fraenkel-Mostowski models

Abstract: Fraenkel-Mostowski models were introduced nearly a century ago as a method of proving the independence of various choice principles from the rest of set theory. Unfortunately, this only works in set theory with atoms (ZFA). However, many Fraenkel-Mostowski proofs can be translated directly into forcing proofs, so this method is of interest for people working with ZF independence proofs.

I will present the construction and give some examples of basic independence results – including the independence of the Axiom of Choice from ZFA and ZF.

Monday, September 9 from 3 to 4pm
Room: Mathematics lunch room
Title: Extracting Measures from Strong Partition Cardinals

Abstract: We say that a cardinal $\kappa$ has the strong partition property if $\kappa\rightarrow(\kappa)^{\kappa}$. A result due to Erdös and Rado tells us that strong partition cardinals do not exist if we take choice. However, in the absence of choice, there is a very natural theory in which strong partition cardinals pop up, namely ZF + AD (Zermelo-Fraenkel set theory with the Axiom of Determinacy). In particular, it turns out that $\aleph_1$ is a strong partition cardinal in ZF + AD. In this talk, I will show how one may extract a measure on strong partition cardinals, using $\aleph_1$ as an example. This talk is expository in nature, and the proofs are entirely combinatorial. Time permitting, I will further discuss the link between determinacy and strong partition cardinals.

Spring 2013

Organizer: Samuel Coskey

Thursday, May 2 from 1:30 to 2:30pm
Room: B-309
Title: Borel determinacy

Abstract: We have seen various determinacy results throughout the semester, all of which have required large cardinal hypotheses. So, it seems natural to see how much determinacy we can get in ZFC. It turns out that we can get quite a bit of determinacy out of the standard axioms, and in fact we see that most sets that the analyst would encounter on a regular basis are determined. With that said, we will go through a high level sketch of Martin’s inductive proof of Borel Determinacy. Our interest here is in the result itself, the elegance of the proof, and the machinery of covering games and unravelings. Time permitting, I will mention applications of Borel Determinacy, as well as other uses of covering games.

Thursday, April 25 from 1:30 to 2:30pm
Room: B-309
Speaker: Marion Scheepers
Title: Embedding partially ordered sets into $(^{\omega}\omega,\prec)$, part II

Thursday, April 18
(no seminar)

Thursday, April 11 from 1:30 to 2:30pm
Room: B-309
Speaker: Marion Scheepers
Title: Embedding partially ordered sets into $(^{\omega}\omega,\prec)$.

Abstract: In previous talks this semester we learned about some universal objects for separable metric spaces, and we learned about cardinal characteristics of the continuum. In this talk we will consider some of these prior ideas and concepts in the context of ordered sets.

Thursday, April 4 from 1:30 to 2:30pm
Room: B-309
Speaker: Samuel Coskey
Title: The Borel ordering on cardinal characteristics

Abstract: Many inequalities between cardinal characteristics of the continuum can be proved in a categorical manner. One simply has to exhibit a certain transformation between the two cardinal invariants called a Tukey map. In several applications, the existence of a Tukey map isn’t enough; one must also know the map can be chosen to be definable. For example, although it is easy to show the pseudo-intersection number $\mathfrak p$ lies below the (un)bounding number $\mathfrak b$, it is not clear if the two cardinals are connected by a definable map. In this talk I’ll introduce the core concept, give some examples, and then consider this slightly more challenging question.

Thursday, March 21 from 1:30 to 2:30pm
Room: B-309
Speaker: Randall Holmes
Title: Of the Urysohn Space $U$, Space-Filling Curves, and Isometric Embeddings of $U$ in $C[0,1]$

Abstract: Dr Coskey recently discussed the universal separable metric space of Urysohn. If you missed his talk, do not fear to come to mine: the essential properties of this space $U$ that are needed for my development will be reviewed in my talk. It is also known that $C[0,1]$, the space of continuous functions on the unit interval with the sup metric, is a universal separable metric space. I will prove this result in my talk: the proof uses space-filling curves. $U$ and $C[0,1]$ are both universal separable metric spaces up to isometry. Of course, since both are universal, $U$ can be embedded isometrically into $C[0,1]$. Sierpinski asked about this (to all appearances rather casually): of course $U$ can be embedded into $C[0,1]$ using his general method involving space-filling curves, but can it be embedded in some other way? I answered this question, and the answer is No. Embeddings of $U$ in $C[0,1]$ always involve space-filling curves in a certain sense. Further, the structure of an isometric copy of $U$ in $C[0,1]$ containing the zero function is exactly determined: if one of the points of $U$ is designated as 0, the norm of any linear combination of points of $U$ when isometrically embedded into $C[0,1]$ (and so into any Banach space) is as large as the distances among the points (and the point designated as 0) permits, and so uniquely determined. Any metric copy of U in a Banach space containing 0 has a uniquely determined Banach space as its linear closure! This is not true for familiar spaces!

Wednesday, March 13 from 4:30 to 5:30pm
Room: MG-226
Title: Analytic Determinacy from a measurable cardinal, II

Thursday, March 7 from 1:30 to 2:30pm
Room: B-309
Title: Analytic Determinacy from a measurable cardinal

Thursday, February 28 from 1:30 to 2:30pm
Room: B-309
Speaker: Samuel Coskey
Title: Urysohn spaces

Abstract: Is there a single separable metric space which contains all the others? This question was answered in the 1920’s by Banach and Mazur, who showed that $C[0,1]$ is such a space. But around the same time Urysohn gave another example (now called Urysohn space $U$) which additionally exhibits strong symmetry properties. Recently Urysohn’s construction has found numerous generalizations and applications. I’ll give a (modern) presentation of the construction, and briefly mention a couple of these recent results.

Thursday, February 21 from 1:30 to 2:30pm
Room: B-309
Speaker: Andrés Caicedo
Title: Finitary mathematics

Abstract: $\mathsf{ZF}_{\mathsf{fin}}$ is the standard formalization of finitary mathematics; it replaces the axiom of infinity in $\mathsf{ZF}$ with its negation. This theory is bi-interpretable with $\mathsf{PA}$, Peano Arithmetic.

This is an expository talk that explains how this bi-interpretability (going back to Ackermann) works, and highlights some of the subtleties that appear along the way.

Thursday, February 14
(no seminar)

Thursday, February 7 from 1:30 to 2:30pm
Room: B-309
Speaker: Andrés Caicedo
Title: Determinacy from large cardinals: an overview II

Abstract: (This talk continues the theme of last week’s talk.) We recall the definition of Woodin cardinals, and explain why they play a key role in proofs of determinacy.

Wednesday, January 30 from 4:30 to 5:30pm
Room: Math conference and break room
Speaker: Andrés Caicedo
Title: Determinacy from large cardinals: an overview

Abstract: A game is determined when one of the players has a winning strategy. Determinacy is the claim that all perfect information ω-length two-player games on integers are determined. This is a key assumption in modern set theory.

We present a high level sketch of how the presence of large cardinals in the universe implies that determinacy holds in the model L(R). There won’t be enough resolution in our microscope to understand all details, but hopefully there will be enough to make sense of the main ingredients of the argument.