Set Theory Seminar

Fall 2017

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

September 1
Joel David Hamkins, CUNY
The inner-model and ground-model reflection principles
The inner model reflection principle asserts that whenever a statement $\varphi(a)$ in the first-order language of set theory is true in the set-theoretic universe $V$, then it is also true in a proper inner model $W\subsetneq V$. A stronger principle, the ground-model reflection principle, asserts that any such $\varphi(a)$ true in $V$ is also true in some nontrivial ground model of the universe with respect to set forcing. Both of these principles, expressing a form of width-reflection in constrast to the usual height-reflection, are equiconsistent with ZFC and an outright consequence of the existence of sufficient large cardinals, as well as a consequence (in lightface form) of the maximality principle. This is joint work with Neil Barton, Andrés Eduardo Caicedo, Gunter Fuchs, myself and Jonas Reitz.

September 8
Brent Cody, Virginia Commonwealth University
The weakly compact reflection principle and orders of weak compactness
There is is a strong analogy between stationary sets and weakly compact sets. However, by a theorem of Kunen there are models in which non-weakly compact sets can become weakly compact after forcing, whereas nonstationary sets can never be forced to become stationary. Thus, proofs about the ideal of non-weakly compact sets are often substationally different than their counterparts for the nonstationary ideal. Many questions whose analogues have been answered for the nonstationary ideal remain open for the weakly compact ideal, and higher order $\Pi^1_n$-indescribability ideals. This talk will survey what is known in this area and will include a discussion of some recent results on the weakly compact reflection principle, which is a generalization of a certain stationary reflection principle. We say that the weakly compact reflection principle holds at $\kappa$ and write $\text{Refl}_{\text{wc}}(\kappa)$ if and only if $\kappa$ is a weakly compact cardinal and every weakly compact subset of $\kappa$ has a weakly compact proper initial segment. It is easy to see that the weakly compact reflection principle $\text{Refl}_{\text{wc}}(\kappa)$ implies that $\kappa$ is an $\omega$-weakly compact cardinal. We will prove the consistency of the existence of an $\omega$-weakly compact cardinal $\kappa$ at which the weakly compact reflection principle $\text{Refl}_{\text{wc}}(\kappa)$ fails, relative to the existence of an $\omega$-weakly compact cardinal; our proof uses a forcing which adds a non-reflecting weakly compact set and preserves all weakly compact cardinals. We will also discuss new joint work with Hiroshi Sakai, in which we prove that the weakly compact reflection principle can hold at the least $\omega$-weakly compact cardinal, and thus the weakly compact reflection principle need not imply that $\kappa$ is $(\omega+1)$-weakly compact. Along the way we generalize the well-known result which states that for regular $\kappa$, after $\kappa$-c.c. forcing the nonstationary ideal of the extension equals the ideal generated by the ground model nonstationary ideal. Our generalization states that if $\kappa$ is weakly compact then after a ‘typical’ Easton-support iteration of length $\kappa$ the weakly compact ideal of the extension equals the ideal generated by the ground model weakly compact ideal.

September 15
No seminar because of Simon Thomas: the first 60 years birthday conference.

September 22
No seminar because of CUNY holidays.

September 29
No seminar because of CUNY holidays.

October 6
Paul Ellis, Manhattanville College
Cycle Reversions and Dichromatic Number in (Infinite) Tournaments
The dichomatic number for a digraph is the least number of acyclic subgraphs needed to cover the graph. In 2005, Pierre Charbit showed that by iterating the operation {{select a directed cycle, and reverse the direction of each arc in it}} that the dichromatic number in any finite digraph can be lowered to 2. This is optimal, as a single directed cycle will always have dichromatic number 2. Recently, Daniel Soukup and I showed that the same is true for infinite tournaments of any cardinality, and in fact, we proved this by induction. Along the way to proving this, we uncovered some nice structural facts about infinite digraphs that we think are of more general interest. While this talk will be mostly graph theoretic in flavor, we did need to put on our set theory glasses to distinguish between the singular and regular cases in the induction. I should note that the question remains open for arbitrary inifinite digraphs, even those of countable cardinality.

October 13
Kameryn Williams, CUNY
The exact strength of the class forcing theorem
Gödel–Bernays set theory $\mathsf{GBC}$ proves that sufficiently nice (i.e. pretame) class forcings satisfy the forcing theorem—that is, these forcing notions $\mathbb P$ admit forcing relations $\Vdash_\mathbb{P}$ satisfying the recursive definition of the forcing relation. It follows that statements true in the corresponding forcing extensions are forced and forced statements are true. But there are class forcings for which having their forcing relation exceeds $\mathsf{GBC}$ in consistency strength. So $\mathsf{GBC}$ does not prove the forcing theorem for all class forcings. This is in contrast to the well-known case of set forcing, where $\mathsf{ZFC}$ proves the forcing theorem for all set forcings. On the other hand, stronger second-order set theories such as Kelley–Morse set theory $\mathsf{KM}$ prove the forcing theorem for all class forcings, providing an upper bound. What is the exact strength of the class forcing theorem?

I will show that, over $\mathsf{GBC}$, the forcing theorem for all class forcings is equivalent to $\mathsf{ETR}_\mathrm{Ord}$ the principle of elementary transfinite recursion for recursions of height $\mathrm{Ord}$. This is equivalent to the existence of $\mathrm{Ord}$-iterated truth predicates for first-order truth relative to any class parameter; which is in turn equivalent to the existence of truth predicates for the infinitary languages $\mathcal L_{\mathrm{Ord}, \omega}(\in, A)$ allowing any class parameter A. This situates the class forcing theorem precisely in the hierarchy of theories between $\mathsf{GBC}$ and $\mathsf{KM}$.

This is joint work with Victoria Gitman, Joel Hamkins, Peter Holy, and Philipp Schlict.

October 20
Victoria Gitman, CUNY
Filter games and Ramsey-like cardinals
Peter Holy and Philipp Schlicht recently introduced a robust hierarchy of Ramsey-like cardinals $\kappa$ using games in which player I plays an increasing sequence of $\kappa$-models and player II responds by playing an increasing sequence of $M$-ultrafilters for some cardinal $\alpha\leq\kappa$-many steps, with player II winning if she is able to continue finding the required filters. The entire hierarchy sits below a measurable cardinal and intertwines with Ramsey cardinals, as well as the Ramsey-like cardinals I introduced earlier. The cardinals in the hierarchy can also be defined by the existence of the kinds of elementary embeddings characterizing Ramsey cardinals and other cardinals in that neighborhood. I will discuss results about the properties of the new hierarchy and the filter games due mainly to Holy and Schlicht, with a few of my own.
An extended abstract can be found here.

October 27
No seminar because of MAMLS Logic Friday.

November 3
Juan Aguilera, Vienna University of Technology
Sigma-projective games
We prove, from hypotheses weaker than the existence of infinitely many Woodin cardinals, that infinite games with payoff in the smallest sigma-algebra containing all open sets of reals and closed under continuous images, are determined.

November 10
Corey Switzer, CUNY
A Cichoń Diagram for Relative Degrees of Constructibility
Following a line of research initiated by Brendle, Brooke-Taylor, Ng and Nies in the context of computability theory, we define and investigate a version of the Cichoń Diagram where, instead of considering the least cardinality of sets with prescribed combinatorial properties, we look at degrees of constructibility relative to a fixed inner model $W$ adding reals with prescribed properties. While many analogies hold with the classical theory, by focusing the attention to the reals, we obtain a theory that is somewhat simpler and more robust. Along the way we present some new results about forcings adding reals. Many questions remain open which we hope to discuss as well.
This will be speaker’s oral exam.

November 17
Note: Carolin Antos had to cancel her visit due to illness.

Gunter Fuchs, CUNY
Determining the extent of weak Todorcevic square principles by reflecting diagonally
Using a new principle of simultaneous reflection of sequences of stationary sets of ordinals which I dubbed diagonal reflection, I precisely determine the effects of the subcomplete forcing axiom on weak forms of Todorcevic’s square principles. These are the two cardinal versions of his version of square, postulating the existence of a threadless coherent sequence of sets of clubs of length the first cardinal, each of which has size at most the second cardinal. I will also sketch how to determine the effects of the proper forcing axiom on these principles.

December 1
Sandra Uhlenbrock, Kurt Gödel Research Center
Canonical inner models and their HODs
An essential question regarding the theory of inner models is the analysis of the class of all hereditarily ordinal definable sets $\operatorname{HOD}$ inside various inner models $M$ of the set theoretic universe $V$ under appropriate determinacy hypotheses. Examples for such inner models $M$ are $L(\mathbb{R})$, $L[x]$ and $M_n(x)$. Woodin showed that under determinacy hypotheses these models of the form $\operatorname{HOD}^M$ contain large cardinals, which motivates the question whether they are fine-structural as for example the models $L(\mathbb{R})$, $L[x]$ and $M_n(x)$ are. A positive answer to this question would yield that they are models of $\operatorname{CH}, \Diamond$, and other combinatorial principles.

The first model which was analyzed in this sense was $\operatorname{HOD}^{L(\mathbb{R})}$ under the assumption that every set of reals in $L(\mathbb{R})$ is determined. In the 1990’s Steel and Woodin were able to show that $\operatorname{HOD}^{L(\mathbb{R})} = L[M_\infty, \Lambda]$, where $M_\infty$ is a direct limit of iterates of the canonical mouse $M_\omega$ and $\Lambda$ is a partial iteration strategy for $M_\infty$. Moreover Woodin obtained a similar result for the model $\operatorname{HOD}^{L[x,G]}$ assuming $\Delta^1_2$ determinacy, where $x$ is a real of sufficiently high Turing degree, $G$ is $\operatorname{Col}(\omega, {<}\kappa_x)$-generic over $L[x]$ and $\kappa_x$ is the least inaccessible cardinal in $L[x]$.

In this talk I will give an overview of these results (including some background on inner model theory) and outline how they can be extended to the model $\operatorname{HOD}^{M_n(x,g)}$ assuming $\boldsymbol\Pi^1_{n+2}$ determinacy, where $x$ again is a real of sufficiently high Turing degree, $g$ is $\operatorname{Col}(\omega, {<}\kappa_x)$-generic over $M_n(x)$ and $\kappa_x$ is the least inaccessible cardinal in $M_n(x)$.

This is joint work with Grigor Sargsyan.

December 8
Ruizhi Yang, Fudan University
On unary union
Joel David Hamkins and Makoto Kikuchi (2016) provided a complete axiomatization for the structure $(V,\subset)$, where $\subset$ is the inclusion relation defined in a model $(V,\in)$ of set theory.

In this talk, we look at other reducts of models of set theory, especially $(V,\textstyle\bigcup)$, where $\textstyle\bigcup$ is the unary union operation defined in $(V,\in)$ as usual. We will provide a computable set of axioms for the theory of $(V,\textstyle\bigcup)$ and show that it is complete by a quantifier-elimination argument. The key observation is that the number of covers of a set $x$, namely the size of $\big\{y\bigm|\textstyle\bigcup y = x\big\}$ is determined only by the size of $x$, furthermore, the number of covers of a certain size is also determined by the size of the covered set. For the finite cases, we have a function $c$ recursively defined as follow.
\[
c(n,m)=\binom{2^{n}}{m}-\sum_{i=1}^n\binom{n}{i}\cdot c(n-i,m).
\]
For $n,m\in\mathbb{N}$, $c(n,m)$ returns the number of covers of $n$ whose sizes are exactly $m$.

The unary union reducts from well-founded countable ZFC models are all isomorphic to each other. However, they are neither computably-saturated models nor prime models.

This is joint work with Joel David Hamkins.