Set Theory Seminar

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.

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.

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.

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.