NYLogic

I will talk about connections between three properties of a forcing class $\Gamma$: the most well-known one is the bounded forcing axiom at $(\omega_1,\lambda)$ for this class, introduced by Goldstern and Shelah for proper forcing. The second property is a two cardinal version of $\Sigma^1_1$ absoluteness, at $(\omega_1,\lambda)$, and the third is the preservation of Aronszajn trees of height $\omega_1$ and width $\lambda$. Under the assumption that $\lambda=\lambda^\omega$ and forcings in $\Gamma$ don't add countable subsets of $\lambda$, I will show that these properties are equivalent. The class of forcing notions satisfying Jensen's property of subcompleteness is a canonical class that does not add reals and it is a corollary of the above result that the bounded forcing axiom for subcomplete forcing at $(\omega_1,2^\omega)$ is equivalent to the preservation of Aronszajn trees of height $\omega_1$ and width $2^\omega$. This generalizes a prior result, obtained jointly with Kaethe Minden. I will discuss some further results, as well as some open questions, on the preservation of Aronszajn trees of height $\omega_1$ and other widths.

This is a continuation of a talk from last semester.

A model $(M, < ,\ldots)$ is said to be $\kappa$-like if $|M| = \kappa$ but for all $a \in M$, $|\{x \in M \mid x < a\}| < \kappa$. Based on the paper, the theory of $\kappa$-like models of arithmetic by R. Kaye, we will identify some axiom schemes true in such models of $I\Delta_0$ and investigate their interesting properties. ​

Several years ago I introduced the Diagonal Reflection Principle (DRP), a maximal form of simultaneous stationary reflection. Roughly, DRP asserts that for all regular $\theta \ge \omega_2$, there are stationarily many sets $W$ of size $\omega_1$ such that every stationary element of $W$ reflects to $W$ (i.e. if $S \subset [\theta]^\omega$ is stationary and $S \in W$, then $S \cap [W \cap \theta]^\omega$ is stationary). In that paper I showed that that $\text{MM}^{+\omega_1}$--even just the '$+\omega_1$' version of the forcing axiom for $\sigma$-closed forcings--implies DRP. In this talk I will prove that MM, even its technical strengthening $\text{MM}^{+\omega}$, does NOT imply DRP, contrary to my initial expectations. This is joint work with Hiroshi Sakai.

In this talk, I will discuss the relative complexity of the isomorphism and bi-embeddability relations for various classes of countable abelian groups from the point of view of descriptive set theory. This is joint work with Filippo Calderoni.

A virtual large cardinal is (usually) the critical point of a generic elementary embedding from a rank-initial segment of the universe into a transitive $M\subset V$, as introduced by Gitman and Schindler (2018). A notable feature is that all virtual large cardinals are consistent with $V=L$, and they've proven useful in characterising several properties in descriptive set theory. We'll work with the virtually $\theta$-measurable, $\theta$-strong and $\theta$-supercompact cardinals, where the $\theta$ in particular indicates that the generic embeddings have $H_\theta^V$ as domain, and investigate how these level-by-level virtual large cardinals relate both to each other and to the existence of winning strategies in certain games. This is work in progress and joint with Philipp Schlicht.

In 1959, MacDowell and Specker proved that every model of Peano Arithmetic has an elementary end extension. Since then, the theorem has been a constant source of new results, as well as a subject of extensions and generalizations. I will give a quick survey of those developments, including some very recent results.

A sequence of random variables is exchangeable when its distribution is invariant to reorderings of the sequence. This notion of symmetry and its generalizations to arrays, graphs, and other structures, arise in many statistical contexts. Representation theorems due to de Finetti, Aldous-Hoover, and others, describe how exchangeable structures can be written in a way that exposes conditional independence, which is often important for computational purposes. However, these results are not obviously computable. We will present several positive and negative results examining the computable content of these theorems, and will also describe several implications for the design of probabilistic programming languages. Joint work with Nathanael Ackerman, Jeremy Avigad, Daniel Roy, and Jason Rute.​

In their 2004 seminal paper, 'Fraisse Limits, Ramsey Theory, and Topological Dynamics of Automorphism Groups,' Kechris, Pestov, and Todorcevic, tied together the fields of model theory, Ramsey theory, descriptive set theory, and topological dynamics, via the concept of homogeneity. A key tool used is a combinatorial concept called finite constraint. We will show that a class of graphs called metrically homogeneous graphs, of interest to model theorists and combinatorialists, is finitely constrained, and we show how this is used to derive a whole host of topological and topological dynamical consequences.

From non-trivial elementary embeddings $j_{1}\dots j_{r}:V_{\lambda}\rightarrow V_{\lambda}$, we obtain a sequence of polynomials $(p_{n}(x_{1},\dots,x_{r}))_{n\in\omega}$ that satisfies the infinite product $$\prod_{k=0}^{\infty}p_{k}(x_{1},\dots,x_{r})=\frac{1}{1-(x_{1}+\dots+x_{r})}.$$ From this infinite product, we deduce lower bounds of the cardinality of $|\langle j_{1},...,j_{r}\rangle/\equiv^{\alpha}|$ using analysis and analytic number theoretic techniques. Computer calculations that search for Laver-like algebras give some empirical evidence that these lower bounds cannot be greatly improved.

We will present the following striking theorem, due to Towsner: If $(M, \mathcal X) \models \mathsf{RCA}_0 + I\Sigma_n$ is countable, then for any (!!!) $W \subseteq M$ there is a countable expansion with the same first order part, $(M, \mathcal Y)$, where $\mathcal X \subseteq \mathcal Y$ so that $(M, \mathcal Y) \models \mathsf{RCA}_0 + I\Sigma_n$ and the set $W$ is $\Delta_{n+1}$ definable. Time permitting, we will then sketch some nice applications of this theorem to the study of conservative extensions of fragments of arithmetic.