CUNY Graduate Center
Hybrid (email Victoria Gitman for meeting id)
Room 3207
Fridays 11:00am-12:30pm
Organized by Gunter Fuchs and Victoria Gitman
Calendar
September 6
11:00am NY time
Hybrid (email Victoria Gitman for meeting id)
Room: 3207
Corey Switzer
Kurt Gödel Research Center
Reflecting Ordinals and Forcing
Abstract
Let $n \lt \omega$ and $\Gamma$ either $\Pi$ or $\Sigma$. An ordinal $\alpha$ is called $\Gamma^1_n$-reflecting if for each $\beta \lt\alpha$ and each $\Gamma^1_n$-formula $\varphi$ if $L_\alpha \models \varphi(\beta)$ then there is a $\gamma \in (\beta, \alpha)$ so that $L_\gamma \models \varphi(\beta)$ where here $\models$ refers to full second order logic. The least $\Sigma^1_n$-reflecting ordinal is called $\sigma^1_n$ and the least $\Pi^1_n$-ordinal is called $\pi^1_n$. These ordinals provably exist and are countable (for all $n \lt \omega$). They arise naturally in proof theory, particularly in calibrating consistency strength of strong arithmetics and weak set theories. Moreover, surprisingly, their relation to one another relies heavily on the background set theory. If $V=L$ then for all $n \lt \omega$ we have $\sigma^1_{n+3} \lt \pi^1_{n+3}$ (due to Cutland) while under PD for all $n \lt \omega$ we have $\sigma^1_n \lt \pi^1_n$ if and only if $n$ is even (due to Kechris).
Surprisingly nothing was known about these ordinals in any model which satisfies neither $V=L$ nor PD. In this talk I will sketch some recent results which aim at rectifying this. In particular we will show that in any generic extension by any number of Cohen or Random reals, a Sacks, Miller or Laver real, or any lightface, weakly homogeneous Borel ccc forcing notion agrees with $L$ about which ordinals are $\Gamma^1_n$-reflecting (for any $n$ and $\Gamma$). Meanwhile, in the generic extension by collapsing $\omega_1$ many interesting things happen, not least amongst them that $\sigma^1_n$ and $\pi^1_n$ are increased - yet still below $\omega_1^L$ for $n > 2$. Along the way we will discuss the plethora of open problems in this area. This is joint work with Juan Aguilera.
Video
September 27
11:00am NY time
Hybrid (email Victoria Gitman for meeting id)
Room: 3207
Takashi Yamazoe
Kobe University
Cichoń's maximum with the uniformity and the covering of the $\sigma$-ideal $\mathcal{E}$ generated by closed null sets
Abstract
Let $\mathcal{E}$ denote the $\sigma$-ideal generated by closed null sets on $\mathbb{R}$. We show that the uniformity and the covering of $\mathcal{E}$ can be added to Cichoń's maximum with distinct values, more specifically, it is consistent that $\aleph_1\lt\mathrm{add}(\mathcal{N})\lt\mathrm{cov}(\mathcal{N})\lt\mathfrak{b}\lt\mathrm{non}(\mathcal{E})\lt\mathrm{non}(\mathcal{M})\lt\mathrm{cov}(\mathcal{M})\lt\mathrm{cov}(\mathcal{E})\lt\mathfrak{d}\lt\mathrm{non}(\mathcal{N})\lt\mathrm{cof}(\mathcal{N})\lt2^{\aleph_0}$ holds.
Video
October 18
11:00am NY time
Hybrid (email Victoria Gitman for meeting id)
Room: 3207
Hanul Jeon
Cornell University
On a cofinal Reinhardt embedding without Powerset
Abstract
Reinhardt embedding is an elementary embedding from $V$ to $V$ itself, whose existence was refuted under the Axiom of Choice by Kunen's famous theorem. There were attempts to get a consistent version of a Reinhardt embedding, and dropping the Axiom of Powerset is one possibility. Richard Matthews showed that $\mathsf{ZFC} + \mathrm{I}_1$ proves $\mathsf{ZFC}$ without Powerset is consistent with a Reinhardt embedding, but the embedding $j\colon V\to V$ in the Matthews' model does not satisfy the cofinality (i.e., for every set $a$ there is $b$ such that $a\in j(b)$). In this talk, I will show from $\mathsf{ZFC} + \mathrm{I}_0$ that $\mathsf{ZFC}$ without Powerset is consistent with a cofinal Reinhardt embedding.
Video
October 25
11:00am NY time
Hybrid (email Victoria Gitman for meeting id)
Room: 3207
Stefan Geschke
University of Hamburg
TBA
Abstract
November 15
11:00am NY time
Hybrid (email Victoria Gitman for meeting id)
Room: 3207
Philipp Schlicht
Kurt Gödel Research Center
TBA
Abstract
November 22
11:00am NY time
Hybrid (email Victoria Gitman for meeting id)
Room: 3207
Alejandro Poveda
Harvard University
TBA
Abstract
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