# Set Theory Seminar

Spring 2018

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.
TBA

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
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
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 20
Michał Tomasz Godziszewski, University of Warsaw
TBA

April 27
Joel David Hamkins
TBA

May 4
Jonas Reitz, CUNY
TBA

May 11
TBA

May 18
Miha Habič, Charles University
TBA

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.