Beijing Institute of Mathematical Sciences and Applications

We consider the following version of the Entscheidungsproblem: given an infinitary propositional formula $\phi$ and a class of forcing notions $\mathcal{K}$, when can we add a satisfying assignment of $\phi$ by a forcing in $\mathcal{K}$. We are mainly interested in the case $\mathcal{K}$ is the class of proper, semiproper, or stationary set preserving forcings. For each of these classes, given a formula $\phi$, we formulate a two person infinite game $G_{\mathcal{K},\phi}$ such that the desired poset exists if and only if player 2 has a winning strategy in this game. We also give sufficient conditions for the existence of such strategies. As an application we sketch a somewhat different proof of the Aspero-Schindler theorem stating that $\mathsf{MM}^{++}$ implies Woodin’s axiom (*). This is joint work with my PhD student Obrad Kasum.

The game of Cops and Robbers on graphs is a well-established model for studying pursuit-evasion scenarios. In this talk, I introduce a modal and epistemic logic perspective on the game, focusing on situations where players have imperfect information. We present the Epistemic Logic of Cops and Robbers (ELCR), a formal framework that captures key elements such as player positions, observational capabilities, and inference. ELCR allows us to model information updates during the game through a novel dynamic operator, providing a logical account of strategic reasoning under uncertainty. We also explore foundational properties of ELCR, including axiomatization and decidability. This represents a first step toward a formal treatment of Cops and Robbers games that accounts for partial information. This is joint work with Dazhu Li and Sujata Ghosh.

A coloring $e: [\omega_1]^2\rightarrow 2$ is ccc if every uncountable family of 0-homogeneous finite sets has two sets whose union is 0-homogeneous. $\mathscr{K}_n$ is the assertion that every ccc poset has property $K_n$. I will show that Martin's axiom for ccc colorings is strictly weaker than MA by constructing a model of Martin's axiom for ccc colorings while some ccc poset is not powerfully ccc. On the other hand, $\mathscr{K}_3$ + $\mathfrak{p}>\omega_1$ implies MA$_{\omega_1}$.

Since Gale and Shapley, economists have studied stable outcomes in two-sided matching markets and the deferred acceptance (Gale–Shapley) algorithm, which finds a stable outcome. The Lone Wolf Theorem illustrates a desirable property of the set of stable outcomes, namely that the set of unmatched agents does not depend on the choice of stable outcome. Classical matching theory has assumed that the set of agents is finite. In this talk, we generalize the Lone wolf Theorem to the infinite setting.

There have been many results showing in specific cases that determinacy is equivalent to its simplest structural consequences. One example from 30 years ago is that hyperprojective determinacy is equivalent to every hyperprojective set has a hyperprojective scale and that there is no uncountable hyperprojective set with a hyperprojective wellordering. This theorem, and its generalizations, are proved through an elaborate induction. For this reason a general equivalence theorem has always seemed out of reach. Our main result is exactly such a theorem. The new ingredients come from an emerging fine structure theory designed to handle the case of "long" extenders. This fine structure theory eliminates the need for an induction.

Club principle is a weak form of diamond principle. Shelah first introduced a forcing poset separating club and diamond principle. Several similar forcing posets were also studied later. In an ongoing project, we introduce another forcing poset separating club and diamond and discuss its new features and some possible application.

In this talk, we provide an intuitive understanding of intuitionistic logic by interpreting it as a logic of knowing how (to prove). The approach is inspired by scattered but related ideas hidden in the vast literature of math, philosophy, CS, and linguistics about intuitionistic logic, which also echoes Heyting's largely forgotten conception of intuitionistic logic as "a logic of knowing". The key technique is to combine the bundled modality we developed in the past decade with the BHK-style interpretation. If time permits, I will demonstrate the use of this epistemic interpretation with applications in inquisitive logic, dependence logic, and deontic logic.

We are focusing on Plato's method of diairesis. This is a method for discovering definitions of concepts that he developed in a series of dialogues, including Phaedrus, Sophist, Statesman, and Philebus. Interestingly, Gregory Cherlin points out that Saharon Shelah's strategy for dividing lines in contemporary model theory is a mathematical version of Plato's "cutting through the middle." John T. Baldwin puts this more specifically: a dividing line is a property that must be both a "virtuous property" and its negation. In this talk, we will focus on more fundamental philosophical questions: Why do dividing lines exist? How can a philosopher or mathematician be sure that a dividing line has been discovered? The requirements for candidate dividing lines proposed by Shelah and the strategy of using the "test problem" parallel various actual methods of dividing in Plato's dialogues. In essence, these requirements and strategies are actually intended to ensure that Plato's requirement in the Sophist is to divide according to Form, that is, objective concepts as we understand them.

Many large cardinals are characterized by the existence of ultrafilters and/or elementary embeddings. The equivalence between these two characterizations relies on Łoś theorem of ultrapowers. However, it is known that Łoś theorem is not provable from $\mathsf{ZF}$, so in the absence of the axiom of choice, the two characterizations would not be equivalent. In this talk, we present several definitions of a supercompact cardinal. One definition is given by elementary embeddings on the universe; the others are ultrafilters, and elementary embeddings on large set models. We prove that these definitions are equivalent or equiconsistent under a small fragment of Łoś theorem. Furthermore, this fragment of Łoś theorem is equivalent to Small Violation of Choice under the existence of a hyperhuge cardinal.

Let $\mathrm{Card}$ denote the class of all cardinals. For all cardinals $\mathfrak{a},\mathfrak{b}$, $\mathfrak{a}\leqslant\mathfrak{b}$ means that there is an injection from a set of cardinality $\mathfrak{a}$ into a set of cardinality $\mathfrak{b}$, and $\mathfrak{a}\leqslant^\ast\mathfrak{b}$ means that there is a surjection from a subset of a set of cardinality $\mathfrak{b}$ onto a set of cardinality $\mathfrak{a}$. A doubly ordered set is a triple $\langle P,\preccurlyeq,\preccurlyeq^\ast\rangle$ such that $\preccurlyeq$ is a partial ordering on $P$, $\preccurlyeq^\ast$ is a preordering on $P$, and ${\preccurlyeq}\subseteq{\preccurlyeq^\ast}$. In 1966, Jech proved that for every partially ordered set $\langle P,\preccurlyeq\rangle$, there exists a model of $\mathsf{ZF}$ in which $\langle P,\preccurlyeq\rangle$ can be embedded into $\langle\mathrm{Card},\leqslant\rangle$. We generalize this result by showing that for every doubly ordered set $\langle P,\preccurlyeq,\preccurlyeq^\ast\rangle$, there exists a model of $\mathsf{ZF}$ in which $\langle P,\preccurlyeq,\preccurlyeq^\ast\rangle$ can be embedded into $\langle\mathrm{Card},\leqslant,\leqslant^\ast\rangle$. This is joint work with Wenjie Zhou.

Motivated by the Hrushovski property, we define a notion of S-extension for a metric space and study minimality and coherence of S-extensions. We show that every S-extension can be identified with an algebraic object. We use this algebraic representation to give a complete characterization of all finite minimal S-extensions of a given finite metric space and a complete characterization of all minimal coherent S-extensions. We also define a notion of ultraextensive metric spaces and show that every countable metric space can be extended to a countable ultraextensive metric space. We also show that the isometry group of an infinite ultraextensive metric space has a dense locally finite subgroup. These results are related to a conjecture of Vershik on the isomorphism type of such dense locally finite subgroups.

In this talk, we recall Borel reducibility among equivalence relations first. Then we introduce equivalence relations $E(G)$ induced by Polish groups $G$. Main part of this talk is to precent many rigid results concerning various kinds of Polish groups: non-archimedean, TSI, CLI, $\alpha$-l.m.-unbalanced, abelian, locally compact, Lie groups, and Banach spaces (as additive groups) and so on.

We show that the Axiom of Real Determinacy ($\mathsf{AD}_{\mathbb{R}}$) and the Axiom of Real Blackwell Determinacy ($\mathsf{Bl}$-$\mathsf{AD}_{\mathbb{R}}$) are equivalent in $\mathsf{ZF}+\mathsf{DC}$. While we do not know if $\mathsf{AD}_{\mathbb{R}}$ and $\mathsf{Bl}$-$\mathsf{AD}_{\mathbb{R}}$ are equivalent without assuming $\mathsf{DC}$, we show that $\mathsf{ZF}+\mathsf{AD}_{\mathbb{R}}$ and $\mathsf{ZF}+\mathsf{Bl}$-$\mathsf{AD}_{\mathbb{R}}$ are equiconsistent. This is joint work with W. Hugh Woodin.

In this talk, I will discuss some applications of methods from set theory and model theory to probability theory and statistics. In probability, there are many questions to which descriptive set-theoretical methods are relevant and I will briefly introduce a new result of this kind connected to de Finetti's Theorem. I will also explain how nonstandard analysis provides an excellent setting for discussing optimality of statistical procedures.