Spring 2018


CUNY Graduate Center
Room 6417
Fridays 10:00-11:45am
Organized by Victoria Gitman and Kameryn Williams

February 2
No seminar because of MathFest 2018 conference at the Graduate Center.

February 9
Neil Barton, Kurt Gödel Research Center
Large Cardinals and the Iterative Conception of Set
Large cardinals are seen as some of the most natural and well-motivated axioms of set theory. Often *maximality* considerations are mobilised in favour of consistent large cardinals: Since (it is argued) the axioms assert that the stages go as far as a certain ordinal, and it is part of the iterative conception that the construction be iterated as far as possible, if it is *consistent* to form a particular large cardinal then we should do so. This paper puts pressure on this line of thinking. We argue that since the iterative conception legislates in favour of forming all possible subsets at each additional stage and *then* iterating this as far as possible, what is regarded as 'consistently formable' will depend upon the nature of the subset operation in play. We present a few cases where the *consistency* of a large cardinal axiom comes apart from its *truth* on the basis of *maximality* criteria. Thus there are interpretations of maximality on which large cardinals are consistent but not true, and so maximality of the iterative conception does not clearly legislate in favour of large cardinals. We will even argue that there might be a natural, maximal, and strong version of set theory on which every set is countable!

February 16
Joseph Van Name, CUNY.
Endomorphic Laver tables
The endomorphic Laver tables extend the notion of a classical Laver table to algebraic structures of arbitrary arity and type.

February 23
Shoshana Friedman, CUNY
Large cardinals and HOD
In his work on the HOD conjecture, Woodin isolates the concept of of an inner model $N$ being a weak extender model for $\delta$ is supercompact: $N$ is an inner model of ZFC, and for every $\gamma>\delta$, there is a $\delta$-complete normal fine measure $U$ on $P_\delta(\gamma)$ such that $N\cap P_\delta(\gamma)\in U$ and $U\cap N\in N$. In the case of HOD, when it is a weak extender model for $\delta$ is supercompact that implies that $\delta$ is HOD-supercompact. In this talk I will separate the concepts of supercompactness, supercompactness in HOD and being HOD-supercompact and how these concepts will be affected by the assumption of the HOD hypothesis.

This is joint work with Arthur Apter and Gunter Fuchs.

March 2
Seminar cancelled! (Rescheduled May 25)
Arthur Apter, CUNY
Strong Compactness, Easton Functions, and Indestructibility
I will discuss realizing Easton functions in the presence of non-supercompact strongly compact cardinals and connections with indestructibility. This is part of a joint project with Stamatis Dimopoulos.

March 9
Brent Cody, Virginia Commonwealth University
The weakly compact reflection principle and orders of weak compactness Part II
This is a continuation of my talk from last semester. Many theorems regarding the nonstationary ideal can be generalized to the ideal of non--weakly compact sets. For example, Hellsten showed that under $\text{GCH}$, if $W\subseteq \kappa$ is a weakly compact set then there is a cofinality-preserving forcing extension in which there is a $1$-club $C\subseteq W$ and all weakly compact subsets of $W$ remain weakly compact. I will discuss some recent results in this direction related to the weakly compact reflection principle, which generalize work of Mekler and Shelah on the nonstationary ideal. One can easily observe that if the weakly compact reflection principle holds at $\kappa$ then $\kappa$ must be $\omega$-weakly compact. By developing a forcing to add a non-reflecting weakly compact set, I will prove that the converse can fail: if $\kappa$ is $(\alpha+1)$-weakly compact then there is a forcing extension in which $\kappa$ remains $\alpha$-weakly compact and the weakly compact reflection principle fails at $\kappa$. I will also discuss a proof of a result joint with Hiroshi Sakai: if the weakly compact reflection principle holds at $\kappa$ then there is a forcing extension preserving this in which $\kappa$ is the least $\omega$-weakly compact cardinal. Hence the weakly compact reflection principle at $\kappa$ need not imply that $\kappa$ is $(\omega+1)$-weakly compact.

March 16 Shehzad Ahmed, Ohio University When is pcf(A) well behaved? Recall that, for a set $A$ of regular cardinals, we define $$\operatorname{pcf}(A) := \{ \operatorname{cf}(\prod A/D): D \text{ is an ultrafilter on A}\}.$$ In the case when $A$ is an interval of regular cardinals satisfying $|A|<\min (A)$, we can say quite a bit about how $\operatorname{pcf}(A)$ behaves. For example, we know that $\operatorname{pcf}(\operatorname{pcf}(A))=\operatorname{pcf}(A)$, and that we can get transitive generators for all $\lambda \in \operatorname{pcf}(A)$. We might then as ourselves what we can say if we remove one or both assumptions of these assumptions on $A$. That is, are there other assumptions under which $\operatorname{pcf}(A)$ is well behaved?

Throughout this talk, I will survey some of the literature regarding this question, and discuss a number of important open questions.

March 23
Kaethe Minden, Marlboro College
Infinite Vatican Squares
In this talk I will discuss ongoing work with Matt Ollis and Gage Martin where we consider natural generalizations of Vatican squares from the finite to the infinite. We construct countable D-complete and Vatican squares using Cayley tables of groups. We show that there are Vatican squares of infinite order and that there is an uncountable semi-Vatican square based on R.

March 30
No seminar because of spring break

April 6
Special time: 2:00-4:00pm
Special room: 6494
Kameryn Williams, CUNY
Dissertation defense: The Structure of Models of Second-order Set Theories
This dissertation is a contribution to the project of second-order set theory, which has seen a revival in recent years. The main results concern the structure of possible collections of classes which may be added to a fixed countable model of ZFC to give a model of second-order set theory. I show that this is a rich structure, with every countable partial order embedding into it. I show that given a countable model of ZFC there is never a smallest collection of classes to put on it to get a model of Kelley–Morse set theory. This implies that there is not a least transitive model of KM, in contrast to the well-known Shepherdson–Cohen theorem that there is a least transitive model of ZFC. More generally, no second-order set theory which proves the existence of the least admissible above V can have a least transitive model. On the other hand, ETR—Gödel–Bernays set theory plus the principle of Elementary Transfinite Recursion—does have a least transitive model. As an important tool towards these results I generalize a construction of Marek and Mostowski which shows that every model of KM (plus the Class Collection schema) "unrolls" to a model of a first-order set theory. I calculate the theories of the unrollings for a variety of second-order set theories, going as weak as ETR. This is used to show that being T-realizable, for a broad class of second-order set theories T, goes down to inner models.

April 13
Ryan Utke, CUNY
An Incompleteness in Peano Arithmetic
In 1931, Godel proved his incompleteness theorem, that any recursive axiomatization of arithmetic is incomplete. As a consequence, there exist true statements about the natural numbers which cannot be proven from the usual Peano axioms. However, the first examples of such statements relied on coding metamathematical concepts such as consistency or provability, and for many decades it was unknown whether any 'natural' statements in arithmetic are true but unprovable. In the 1970s, such a statement was given by Paris and Harrington, whose argument we present. In particular, we prove (in ZFC) that a variant of the finite Ramsey theorem is true but implies the consistency of PA, hence cannot be proven in PA.

April 20
Michał Tomasz Godziszewski, University of Warsaw
Set-theoretic independence of existence of some local hidden variable models in the foundations of quantum mechanics.
In 1982 I. Pitowsky gave a construction of local hidden variable models (i.e. descirptions of quantum systems that provide deterministic predictions and are satisfied for spatially separated observables) for the so-called spin-1/2 (and spin-1) particles in quantum mechanics. Specifically, Pitowsky's main result was that under the assumption of the Continuum Hypothesis there exists a spin-1/2 function. The function constructed in the proof of the theorem is non-measurable, making Pitowsky's model not directly subject to famous Bell's theorem, stating that under certain assumptions local hidden variable models are impossible. Since the construction uses CH, the natural question is whether the existence of Pitowsky's functions is provable in ZFC. In 2012, I. Farah and M. Magidor demonstrated that it is actually independent. Namely they proved that: (1) if there exists a $\sigma$-additive extension of the Lebesgue measure to the power-set of the reals (i.e. if the large cardinal axiom known as 'the continuum is a real-valued measurable cardinal' holds), then Pitowsky's models do not exist, and (2) Pitowsky's models do not exist in the random real model. The second result thus shows that the non-existence of Pitowsky's functions is relatively consistent with ZFC. The proofs of these theorem rely on the results by H. Friedman and D.H. Fremlin, specifying that under the assumptions of (1) and (2) Pitowsky's functions have to be Borel-measurable. During the talk I will try to sketch Pitowsky's construction, explain the proofs of the Farah-Magidor theorems, and, if time permits, relate them to some ongoing debates in the quantum foundations and philosophy of set theory.

April 27
Joel David Hamkins
Determinacy for open class games is preserved by forcing
Open class determinacy is the principle of second order set theory asserting of every two-player game of perfect information, with plays coming from a (possibly proper) class $X$ and the winning condition determined by an open subclass of $X^\omega$, that one of the players has a winning strategy. This principle finds itself about midway up the hierarchy of second-order set theories between Gödel-Bernays set theory and Kelley-Morse, a bit stronger than the principle of elementary transfinite recursion ETR, which is equivalent to clopen determinacy, but weaker than GBC+$\Pi^1_1$-comprehension. In this talk, I’ll given an account of my recent joint work with W. Hugh Woodin, proving that open class determinacy is preserved by forcing. A central part of the proof is to show that in any forcing extension of a model of open class determinacy, every well-founded class relation in the extension is ranked by a ground model well-order relation. This work therefore fits into the emerging focus in set theory on the interaction of fundamental principles of second-order set theory with fundamental set theoretic tools, such as forcing. It remains open whether clopen determinacy or equivalently ETR is preserved by set forcing, even in the case of the forcing merely to add a Cohen real.

May 4
Jonas Reitz, CUNY
Cohen Forcing and Inner Models
The most well-known method for adding a subset to a regular cardinal kappa over the universe $V$ is the Cohen partial order ${\rm Add}(\kappa,1)$, whose conditions consist of binary sequences bounded in $\kappa$ and ordered by end extension. Presented with an inner model $W$, however, we can consider an alternative: add a subset to kappa over the universe $V$ using ${\rm Add}(\kappa,1)^W$, the Cohen partial order as defined within $W$. This situation is not unusual in set theory - it arises in the canonical methods for adding subsets to multiple cardinals (in product forcing we always use the poset of the ground model $W$, whereas in iterations we use the poset of the extension), and in many other inner and outer model constructions. As $W$ may have fewer bounded subsets of kappa than $V$, these posets may not be equal. How do they compare?

In this talk I will analyze ${\rm Add}(\kappa,1)^W$ and ${\rm Add}(\kappa,1)^V$ with regards to their relative forcing strength. I will offer a complete characterization of when forcing with ${\rm Add}(\kappa,1)^W$ is (at least) as strong as ${\rm Add}(\kappa,1)^V$, in the sense that forcing with the former poset adds a generic for the latter. I will present partial results in the reverse direction (when is ${\rm Add}(\kappa,1)^V$ as strong as ${\rm Add}(\kappa,1)^W$?), and discuss open questions and applications.

May 11
Alfredo Roque Freire, Universidade Estadual de Campinas
On bi-interpretations in Set Theory that preserve isomorphism
In this presentation I will show that no different set theories extending ZF can be bi-interpretable. This result was achieved by Joel Hamkins and I this February 2018, but it was later discovered that Ali Enayat have already proved the result in 2017. I will also present what I believe to be a solid strengthening of the result: for any two models $\mathcal{M}$ and $\mathcal{N}$ of Zermelo set theory (Z: ZF without replacement), if they are mutually interpretable trough interpretations $I$ and $J$, there is a isomorphism $b$ from $\mathcal{M}$ to ${\mathcal{M}^I}^J$ and replacement is valid in $\mathcal{M}$ for the function $b$, then $\mathcal{M}$ and $\mathcal{N}$ are isomorphic. Along the way I will introduce some definitions, intending to bring clarity to how meaningful this result is; Moreover, I will partially answer the opposite question: what makes it possible for two different theories to be bi-interpretable? The answer to this question will take the form of describing classes of theories for which these property holds.

May 18
Miha Habič, Charles University
Tukey classes of complete ultrafilters
A poset $P$ is Tukey reducible to a poset $Q$ if there is a map $f\colon Q\to P$ which takes cofinal subsets of $Q$ to cofinal subsets of $P$. The classification of all Tukey classes of posets of size continuum is not feasible, but becomes more tractable if we restrict our attention to a particular class of posets. Of particular interest are Tukey reductions between ultrafilters on $\omega$, ordered by inclusion, and it is a long-standing open question whether it is consistent that all nonprincipal ultrafilters are Tukey equivalent. I will give an overview of known results on the topic, and present some new joint work on the parallel question in the case of complete ultrafilters on uncountable cardinals.

May 25
Omer Ben-Naria, University of California at Los Angeles
TBA

May 25
Arthur Apter, CUNY
Strong Compactness, Easton Functions, and Indestructibility
I will discuss realizing Easton functions in the presence of non-supercompact strongly compact cardinals and connections with indestructibility. This is part of a joint project with Stamatis Dimopoulos.