CUNY Graduate Center
Virtual
Organized by Victoria Gitman, Gunter Fuchs, and Arthur Apter
Fall 2021
December 10
The seminar will take place virtually at 2pm US Eastern Standard Time. Please email Victoria Gitman for meeting id.
Eyal Kaplan
Tel Aviv University
Non-stationary support iterations of Prikry forcings and restrictions of ultrapower embeddings to the ground model: Part II
Abstract
Assume that $\mathbb P$ is a forcing notion, $G$ is a generic set for it over the ground model $V$, and a cardinal $\kappa$ is measurable in the generic extension. Let $j$ be an ultrapower embedding, taken in $V[G]$ with a normal measure on $\kappa$. We consider the following questions:
1. Is the restriction of $j$ to $V$ an iterated ultrapower of $V$ (by its measures or extenders)?
2. Is the restriction of $j$ to $V$ definable in $V$?
By a work of Schindler [1], the answer to the first question is affirmative, assuming that there is no inner model with a Woodin Cardinal and $V=K$ is the core model. By a work of Hamkins [2], the answer to the second question is positive for forcing notions which admit a Gap below $\kappa$.
We will address the above questions in the context of nonstationary-support iteration of Prikry forcings below a measurable cardinal $\kappa$. Assuming GCH only in the ground model, we provide a positive answer for the first question, and describe in detail the structure of $j$ restricted to $V$ as an iteration of $V$. The answer to the second question may go either way, depending on the choice of the measures used in the Prikry forcings along the iteration; we will provide a simple sufficient condition for the positive answer. This is a joint work with Moti Gitik.
[1] Ralf Schindler. Iterates of the core model. Journal of Symbolic Logic, pages 241–251, 2006.
[2] Joel David Hamkins. Gap forcing. Israel Journal of Mathematics, 125(1):237–252, 2001.
Video
December 3
The seminar will take place virtually at 2pm US Eastern Standard Time. Please email Victoria Gitman for meeting id.
Eyal Kaplan
Tel Aviv University
Non-stationary support iterations of Prikry forcings and restrictions of ultrapower embeddings to the ground model
Abstract
Assume that $\mathbb P$ is a forcing notion, $G$ is a generic set for it over the ground model $V$, and a cardinal $\kappa$ is measurable in the generic extension. Let $j$ be an ultrapower embedding, taken in $V[G]$ with a normal measure on $\kappa$. We consider the following questions:
1. Is the restriction of $j$ to $V$ an iterated ultrapower of $V$ (by its measures or extenders)?
2. Is the restriction of $j$ to $V$ definable in $V$?
By a work of Schindler [1], the answer to the first question is affirmative, assuming that there is no inner model with a Woodin Cardinal and $V=K$ is the core model. By a work of Hamkins [2], the answer to the second question is positive for forcing notions which admit a Gap below $\kappa$.
We will address the above questions in the context of nonstationary-support iteration of Prikry forcings below a measurable cardinal $\kappa$. Assuming GCH only in the ground model, we provide a positive answer for the first question, and describe in detail the structure of $j$ restricted to $V$ as an iteration of $V$. The answer to the second question may go either way, depending on the choice of the measures used in the Prikry forcings along the iteration; we will provide a simple sufficient condition for the positive answer. This is a joint work with Moti Gitik.
[1] Ralf Schindler. Iterates of the core model. Journal of Symbolic Logic, pages 241–251, 2006.
[2] Joel David Hamkins. Gap forcing. Israel Journal of Mathematics, 125(1):237–252, 2001.
Video
November 19
The seminar will take place virtually at 2pm US Eastern Standard Time. Please email Victoria Gitman for meeting id.
Corey Switzer
University of Vienna
Definable Well Orders and Other Beautiful Pathologies
Abstract
Many sets of reals - well orders of the reals, MAD families, ultrafilters on omega etc - only necessarily exist under the axiom of choice. As such, it has been a perennial topic in descriptive set theory to try to understand when, if ever, such sets can be of low definitional complexity. Large cardinals rule out such the existence of projective well orders, MAD families etc while it's known that if $V=L$ (or even just 'every real is constructible') then there is a $\Delta^1_2$ well order of the reals and $\Pi^1_1$ witnesses to many other extremal sets of reals such as MAD families and ultrafilter bases. Recently a lot of work on the border of combinatorial and descriptive set theory has focused on considering what happens to the definitional complexity of such sets in models in which the reals have a richer structure - for instance when $\mathsf{CH}$ fails and various inequalities between cardinal characteristics is achieved. In this talk I will present a recent advance in this area by exhibiting a model where the continuum is $\aleph_2$, there is a $\Delta^1_3$ well order of the reals, and a $\Pi^1_1$ MAD family, a $\Pi^1_1$ ultrafilter base for a P-point, and a $\Pi^1_1$ maximal independent family, all of size $\aleph_1$. These complexities are best possible for both the type of object and the cardinality hence this may be seen as a maximal model of 'minimal complexity witnesses'. This is joint work with Jeffrey Bergfalk and Vera Fischer.
Video
November 12
The seminar will take place virtually at 1pm US Eastern Standard Time. Please email Victoria Gitman for meeting id.
Tom Benhamou
Tel Aviv University
Intermediate Prikry-type models, quotients, and the Galvin property: Part II
Abstract
We classify intermediate models of Magidor-Radin generic extensions. It turns out that similar to Gitik Kanovei and Koepke's result, every such intermediate model is of the form $V[C]$ where $C$ is a subsequence of the generic club added by the forcing. The proof uses the Galvin property for normal filters to prove that quotients of some Prikry-type forcings are $\kappa^+$-c.c. in the generic extension and therefore do not add fresh subsets to $\kappa^+$. If time permits, we will also present results regarding intermediate models of the Tree-Prikry forcing.
Video Slides
November 5
The seminar will take place virtually at 1pm US Eastern Standard Time. Please email Victoria Gitman for meeting id.
Tom Benhamou
Tel Aviv University
Intermediate Prikry-type models, quotients, and the Galvin property
Abstract
We classify intermediate models of Magidor-Radin generic extensions. It turns out that similar to Gitik Kanovei and Koepke's result, every such intermediate model is of the form $V[C]$ where $C$ is a subsequence of the generic club added by the forcing. The proof uses the Galvin property for normal filters to prove that quotients of some Prikry-type forcings are $\kappa^+$-c.c. in the generic extension and therefore do not add fresh subsets to $\kappa^+$. If time permits, we will also present results regarding intermediate models of the Tree-Prikry forcing.
Video
Slides
October 29
The seminar will take place virtually at 2pm US Eastern Standard Time. Please email Victoria Gitman for meeting id.
Kameryn Williams
Sam Houston State University
Potentialism about classes
Abstract
Set-theoretic potentialism is the view that the universe of sets is never fully completed but is only given potentially. Tools from modal logic have been applied to understand the mathematics of potentialism. In recent work, Neil Barton and I extended this analysis to class-theoretic potentialism, the view that proper classes are given potentially (while the sets may or may not be fixed).
In this talk, I will survey some results from set-theoretic potentialism. After seeing how the tools apply in that context I will then discuss our work in the class-theoretic context.
Video
October 15
The seminar will take place virtually at 2pm US Eastern Standard Time. Please email Victoria Gitman for meeting id.
Yuxin Zhou
University of Florida
Color isosceles triangles countably in $\mathbb{R}^2$ and but not in $\mathbb{R}^3$
Abstract
Let $n>1$ be a natural number, let $\Gamma_n$ be the hypergraph of isosceles triangles in $\mathbb{R}^n$. Under the axiom of choice, the existence of a countable coloring for $\Gamma_n$ is true for every $n$. Without the axiom of choice, the coloring problems will be hard to answer. We often expect the case that the countable chromatic number of one hypergraph doesn't imply the one for another. With an inaccessible cardinal, there is a model of ZF+DC in which $\Gamma_2$ has countable chromatic number while $\Gamma_3$ has uncountable chromatic number. This result is obtained by a balanced forcing over the symmetric Solovay model.
Video
October 8
The seminar will take place virtually at 2pm US Eastern Standard Time. Please email Victoria Gitman for meeting id.
Brent Cody
Virginia Commonwealth University
Higher derived topologies
Abstract
By beginning with the order topology on an ordinal $\delta$, and iteratively declaring more and more derived sets to be open, Bagaria defined the derived topologies $\tau_\xi$ on $\delta$, where $\xi$ is an ordinal. He showed that the non-isolated points in the space $(\delta,\tau_\xi)$ can be characterized using a strong form of iterated simultaneous stationary reflection called $\xi$-s-reflection, which is deeply connected to certain transfinite indescribability properties. However, Bagaria's definitions break for $\xi\geq\delta$ because, under his definitions, the $\delta$-th derived topology $\tau_\delta$ is discrete and no ordinal $\alpha$ can be $\alpha+1$-s-stationary. We will discuss some new work in which we use certain diagonal versions of Bagaria's definitions to extend his results. For example, we introduce the notions of diagonal Cantor derivative and use it to obtain a sequence of derived topologies on a regular $\delta$ that is strictly longer than that of Bagaria's, under certain hypotheses.
Video
October 1
The seminar will take place virtually at 11:30am US Eastern Standard Time. Please email Victoria Gitman for meeting id.
Matteo Viale
University of Torino
Absolute model companionship, forcibility, and the continuum problem: Part II
Abstract
Absolute model companionship (AMC) is a strengthening of model companionship defined as follows: For a theory $T$, $T_{\exists\vee\forall}$ denotes the logical consequences of $T$ which are boolean combinations of universal sentences. $T^*$ is the AMC of $T$ if it is model complete and $T_{\exists\vee\forall}=T^*_{\exists\vee\forall}$. The $\{+,\cdot,0,1\}$-theory $\mathsf{ACF}$ of algebraically closed field is the model companion of the theory of $\mathsf{Fields}$ but not its AMC as $\exists x(x^2+1=0)\in \mathsf{ACF}_{\exists\vee\forall}\setminus \mathsf{Fields}_{\exists\vee\forall}$. Any model complete theory $T$ is the AMC of $T_{\exists\vee\forall}$. We use AMC to study the continuum problem and to gauge the expressive power of forcing. We show that (a definable version of) $2^{\aleph_0}=\aleph_2$ is the unique solution to the continuum problem which can be in the AMC of a partial Morleyization of the $\in$-theory $\mathsf{ZFC}+$there are class many supercompact cardinals. We also show that (assuming large cardinals) forcibility overlaps with the apparently weaker notion of consistency for any mathematical problem $\psi$ expressible as a $\Pi_2$-sentence of a (very large fragment of) third order arithmetic ($\mathsf{CH}$, the Suslin hypothesis, the Whitehead conjecture for free groups are a small sample of such problems $\psi$). Partial Morleyizations can be described as follows: let $\mathsf{Form}_{\tau}$ be the set of first order $\tau$-formulae; for $A\subseteq \mathsf{Form}_\tau$, $\tau_A$ is the expansion of $\tau$ adding atomic relation symbols $R_\phi$ for all formulae $\phi$ in $A$ and $T_{\tau,A}$ is the $\tau_A$-theory asserting that each $\tau$-formula $\phi(\vec{x})\in A$ is logically equivalent to the corresponding atomic formula $R_\phi(\vec{x})$. For a $\tau$-theory $T$ $T+T_{\tau,A}$ is the partial Morleyization of $T$ induced by $A\subseteq \mathsf{Form}_\tau$.
Video
September 24
The seminar will take place virtually at 11:30am US Eastern Standard Time. Please email Victoria Gitman for meeting id.
Matteo Viale
University of Torino
Absolute model companionship, forcibility, and the continuum problem
Abstract
Absolute model companionship (AMC) is a strengthening of model companionship defined as follows: For a theory $T$, $T_{\exists\vee\forall}$ denotes the logical consequences of $T$ which are boolean combinations of universal sentences. $T^*$ is the AMC of $T$ if it is model complete and $T_{\exists\vee\forall}=T^*_{\exists\vee\forall}$. The $\{+,\cdot,0,1\}$-theory $\mathsf{ACF}$ of algebraically closed field is the model companion of the theory of $\mathsf{Fields}$ but not its AMC as $\exists x(x^2+1=0)\in \mathsf{ACF}_{\exists\vee\forall}\setminus \mathsf{Fields}_{\exists\vee\forall}$. Any model complete theory $T$ is the AMC of $T_{\exists\vee\forall}$. We use AMC to study the continuum problem and to gauge the expressive power of forcing. We show that (a definable version of) $2^{\aleph_0}=\aleph_2$ is the unique solution to the continuum problem which can be in the AMC of a partial Morleyization of the $\in$-theory $\mathsf{ZFC}+$there are class many supercompact cardinals. We also show that (assuming large cardinals) forcibility overlaps with the apparently weaker notion of consistency for any mathematical problem $\psi$ expressible as a $\Pi_2$-sentence of a (very large fragment of) third order arithmetic ($\mathsf{CH}$, the Suslin hypothesis, the Whitehead conjecture for free groups are a small sample of such problems $\psi$). Partial Morleyizations can be described as follows: let $\mathsf{Form}_{\tau}$ be the set of first order $\tau$-formulae; for $A\subseteq \mathsf{Form}_\tau$, $\tau_A$ is the expansion of $\tau$ adding atomic relation symbols $R_\phi$ for all formulae $\phi$ in $A$ and $T_{\tau,A}$ is the $\tau_A$-theory asserting that each $\tau$-formula $\phi(\vec{x})\in A$ is logically equivalent to the corresponding atomic formula $R_\phi(\vec{x})$. For a $\tau$-theory $T$ $T+T_{\tau,A}$ is the partial Morleyization of $T$ induced by $A\subseteq \mathsf{Form}_\tau$.
Video
September 3
The seminar will take place virtually at 2pm US Eastern Standard Time. Please email Victoria Gitman for meeting id.
Joan Bagaria
Universitat de Barcelona
Huge Reflection, and beyond
Abstract
We shall present some recent results from a joint work with Philipp Lücke on Structural Reflection at the upper ridges of the large-cardinal hierarchy. In particular, we will introduce a natural form of reflection we call 'Exact Reflection', giving upper and lower bounds for its consistency strength. We will also discuss 'sequential' forms of Exact Reflection, which may be viewed as strong forms of Chang's Conjecture, and which, in the case of infinite sequences, their strength goes beyond the strongest large cardinal principles that are not known to be inconsistent with the Axiom of Choice.
Video