# Stefan Mesken

## Set Theorist

Currently a PhD student at the WWU Münster under supervision of Ralf Schindler. My research interests lie in Inner Model Theory, especially Inner Model Theoretic Geology, see G. Fuchs, R. Schindler. Inner Model Theoretic Geology and G. Sargsyan, R. Schindler. Varsovian Models I.

You can reach me at my email: s (dot) mesken (at) wwu (dot) de

# Current Projects

• Inner Model Theoretic Geology
• Woodin Cardinals in Core Models

# Talks

• Bedrocks in Extender Models, Set Theory Seminar CUNY, USA, March 15th, 2019
• Varsovian Models with more Woodin Cardinals, Research Seminar WWU Münster, Germany, April 17th, April 24th & May 5th, 2018
• Inner Model Theoretic Geology, 4th Münster, Conference on Inner Model Theory, Germany, July 27, 2017
• The Mantle of $L[x]$, Young Researcher’s Seminar Week, Barcelona, Spain, November 7 & 11, 2016
• Set Theoretical Aspects of Category Theory, Research Seminar of the Workgroup Prof. Schneider, Münster, Germany, June 21, 2016
• A Brief Introduction to the Stationary Tower, Young Set Theory Workshop, Copenhagen, Denmark, June 16, 2016

# 2019

## Fine Structure Seminar (Rutgers)

Together with Dan Saattrup Nielsen and Grigor Sargsyan, we give a seminar on the basics of fine structure in preparation of the upcoming The Core Model Induction and Other Inner Model Theoretic Tools meeting in June.

## Descriptive Set Theory Seminar

Continuation of last year's seminar with Andreas Lietz.

# 2018

## Descriptive Set Theory Seminar

Motivated by our recently sparked interest in Core Model Induction, Gabriel Fernandes, Andreas Lietz and I decided to organize an informal weekly seminar on Descriptive Set Theory. Our goal is to introduce our students to the basic concepts of classical descriptive set theory and then, building on the previous fine structure seminar, show up some links to Inner Model Theory.

## Normalizing Iteration Trees

Together with Gabriel Fernandes and Dan Saattrup Nielsen, I'm organizing a weekly, informal seminar on John Steel. Normalizing iteration trees and comparing iteration strategies.

## Fine Structure Seminar

Continuation of last year's weekly seminar on fine structure with Gabriel Fernandes. This year's topics include:

• premice and the Mitchell-Steel indexing,
• iteration trees,
• some large cardinals and their representation via extender sequences,
• genericity iterations via the extender algebra,
• copying constructions,
• the Dodd-Jensen lemma,
• Set Theoretic Geology and
• The Mantle of $L[x]$.

# 2017

## Fine Structure Seminar

Together with Gabriel Fernandes I'm giving an informal, weekly seminar on the basics of fine structure leading to the study of small mice. Some of the topics we've discussed:

• acceptable $\mathcal{J}$-structures,
• iterated projecta and mastercodes,
• $\Sigma^*$-formulae, their interpretation and the correspondence to $\Sigma$-formulae over sound structures,
• extenders and their iterates, fine structural ultrapowers,
• "baby" mice and their comparison.

# Problems in Set Theory

## Girona Meeting 2018

Problem By Solved?
Is there an extender model $L[E]$ such that the mantle of $L[E]$ is not a fully iterable (from the point of view of $L[E]$) hod mouse? Ralf Schindler
A reducible ultrafilter is a countably complete ulterfilter whose ultrapower embedding doesn't factor as a finite linear ultrapower embedding by internal ultrafilters.
Schlutzenberg-Steel: In $L[E]$ the irreducible ultrafilters are exactly the measures on the sequence.
Is it consistent with a $\kappa^{+}$ supercompact cardinal that the Mitchell order restricted to irreducible ultrafilters is a well-ordering?
In $L[E]$, if there is a normal measure concentrating on $\alpha$ which are $\alpha^{+}$-supercompact then there is a normal measure which is not on the sequence.
Reference: Fine structure at the finite levels of supercompactness.
Gabriel Goldberg
Let $(M, \Sigma)$ be a mouse pair, $S$ an iteration tree of successor length on $M$ (not necessarily by $\Sigma$), with a nice $m$-strategy $\Lambda$. Is $\Lambda = \Psi^{*}_{S}$ for some $\Psi$ such that $(M, \Psi)$ is a mouse pair?
Analogous question for type $2$ $m$-trees?
Benny Siskind
What is the consistencty strength of the determincacy of the least $\sigma$-algebra containing all the projective sets?
Lower bound: $\mathrm{PD}$.
Upper bound: Let $N^{\#}_{\omega + 1}(x)$ be the least active $x$-mouse $P$ with a Woodin cardinal $\delta$ such that $\forall n < \omega \colon M^{\#}_{n}(P | \delta) \triangleleft P$. An upper bound is then $\forall x \in \mathbb{R} \colon N^{\#}_{\omega + 1}(x) \text{ exists}$.
Juan Aguilera
What is the consistencty strength of $\mathrm{ZF} + \text{every Suslin set is } \boldsymbol{\Sigma}^{1}_{2}$?
(Add $\mathrm{DC}$ if you want.)
Upper bound: A generic Vopěnka cardinal.
Lower bound: $\mathrm{ZFC}$
Trevor Wilson
Let $\kappa \ge \aleph_{1}$ be a regular cardinal. Suppose $M$ has a $\kappa + 1$ iteration strategy with strong hull condensation.
Fact: If $\mathbb{P}$ has the $\kappa$-c.c., then this is still true in $V^{\mathbb{P}}$.
For which other forcings is this also true (or false)?
Example (Schindler): If $\kappa = \aleph_{1}$, can have a proper forcing $\mathbb{P}$ s.t. $V^{\mathbb{P}} \models M^{\#}_{1} \text{ is not } \omega_{1} + 1 \text{-iterable }$ but $V \models M^{\#}_{1} \text{ is } \omega_{1} + 1 \text{-iterable}$.
What about $\omega$-closed forcings?
See "Iterability for Stacks" on Farmer's homepage.
Farmer Schlutzenberg
Let $\Gamma$ be a definable class of forcings. The $\Gamma$-mantle is the intersection of all grounds $W$ with $W[g] = V$ for some $g$ which is $\mathbb{P}$ generic oveer $W$ and $\mathbb{P} \in \Gamma^{W}$
• When is the $\Gamma$-mantle a model of $\mathrm{ZF}$? (A sufficient condition for this is that the $\Gamma$-grounds are downward directed. See "Set Theoretic Geology" by Fuchs, Hamkins, Rietz.)
• Can the $\Gamma$-mantle be a model of $\mathrm{ZF} + \mathrm{AD}$?
Fuchs: The answer is "yes": If $\Gamma$ is the lass of all $\sigma$-closed forcings we have that the $\Gamma$-mantle of $L(\mathbb{R})^{\mathrm{Coll}(\omega_1, \mathbb{R})}$ is $L(\mathbb{R})$.
• What about $\Gamma$-mantles in $L[E]$-models?
Consider for example the least $L[E]$ with a Woodin cardinal above a strong cardinal and let $\Gamma$ be the class of $\sigma$-closed forcings.
Stefan Mesken
Suppose $M^{\#}_{1} \le_{T} x$ and work in $L[x]$. A proper class inner model $N$ is $M_{1}$-like if $N \models \text{ I'm } M_{1} + \delta^{N} < \omega_{1}$.
Let $H$ be the result of simultaneously pseudo comparing all such $N$.
Fact: $H = L[\mathcal{M}(\mathcal{T})]$ for any iteration tree $\mathcal{T}$ from that comparison. $\delta^{H} = \omega_{2}$.
Is $H | \omega_{2} = \mathrm{HOD}^{L[x]}| \omega_{2}$?
John Steel
Let $M_{\mathrm{refl}}$ be the least mouse with $\lambda$, a limit of Woodins an $< \lambda$-strong and some $\kappa < \lambda$ reflecting the set of $< \lambda$-strongs, i.e. $\kappa$ is $A$ strong up to $\lambda$ for $A := \{\mu < \lambda \mid \mu \text{ is } < \lambda \text{-strong }\}$. Assume $M_{\mathrm{refl}}$ has a good $\omega_{1} + 1$ iteration strategy.
Does $M_{\mathrm{refl}}$ have a uniquely assigned derived model? E.g. does $\mathbb{D}(M, <) \models \mathrm{LSA}$?
Steel: Let $M_0, M_1$ be $\mathbb{R}$-genericity iterates of $M_{\mathrm{refl}}$, i.e. $M_{\mathrm{refl}} \overset{i_j}{\longrightarrow} M_j$. Let $g_j$ be $\mathrm{Coll}(\omega, < i(\lambda))$-generic over $M_j$ such that $\mathbb{R}^{*}_{g_{j}} = \mathbb{R}^{V}$. Do we have $\mathrm{Hom}^{*}_{g_{0}} = \mathrm{Hom}^{*}_{g_{1}}$?
Does $\mathrm{AD}^{+} + \text{ no long extender}$ imply hod-pair capturing? John Steel
Is there an $L[E]$ model which has an inner model $N$ such that $N[g] = L[E]$, where $g$ is $\mathbb{P}$ generic for some $\mathbb{P} \in N$ such that $N \models \mathbb{P} \text{ is } \sigma \text{-closed (/ distributive)}$? Gabriel Goldberg
Assume $\mathrm{AD}^{L(\mathbb{R})}$. Does $L(\mathbb{R})$ satisfy the "ground axiom"?
We put "ground axiom" in quotation marks here because we aren't sure whether this is necessarily first order expressible over a model of $\mathrm{ZF} + \neg \mathrm{C}$.
Gabriel Goldberg

## Münster Meeting 2017

Problem By Solved?
What is $\mathrm{HOD}^{L(\mathbb{R})}$ in the lbr hierarchy?
Same for $L[x,g]$.
Same under $\mathrm{AD}^{+} + \neg \mathrm{AD}_{\mathbb{R}} + \mathrm{HPC}$ (hod pair capturing)
John Steel
Let $M$ be the least extender model with a $\lambda$ which is a limit of strong cardinals. What is the mantle?
Does it have a bedrock?
Is it the $\mathrm{HOD}$ of the $\mathrm{AD}$-model construction from $M$ by Larson-Sargsyan-Wilson (the least model containing $\mathbb{R}^{*}$, $\mathrm{Hom}^{*}$ in which every set of reals is universally Baire)?
Ralf Schindler
Assume $\kappa$ is a singular, strong limit, $\mathrm{card}(V_{\kappa}) = \kappa$ and $\neg \square_{\kappa}$. Is there a proper class model containing all reals satisfying $\mathrm{AD}^{+} + \mathrm{LSA}$?
Known:
• non-tame mouse (Grigor Sargsyan)
• very likely: using the methods for Trang's proof of $\mathrm{PFA} \implies \mathrm{Con}(\mathrm{AD}_{\mathbb{R}} + \theta \text{ regular})$ one can obtain $\mathrm{AD}_{\mathbb{R}} + \theta \text{ regular}$
Let $\mathcal{M}_{1, \infty}$ be the direct limit of countable interates of $M_1$. Let $A \in \mathcal{M}_{1, \infty}$. Must there be some $n < \omega$ such that either $\{ u_m \mid n \le m < \omega\} \subseteq A$ or $\{ u_m \mid n \le m < \omega\} \cap A = \emptyset$?
Background: We have a characterization of $\mathcal{M}_{1, \infty}$ in terms of descriptive set theory. $\mathcal{M}_{1, \infty} = L_{u_{\omega}}[0_{\Sigma^{1}_{3}}]$, where $0_{\Sigma^{1}_{3}} = \{ (\phi, \alpha) \in \text{fml} \times u_{\omega} \mid \exists w \phi(w) \wedge w \text{ codes } \alpha \text{ in the sharp coding} \}$.
Best knowledge thus far: For each $n < \omega$: $u_n \in \mathrm{ran}(\pi_{M_1, \mathcal{M}_{1, \infty}})$.
References:
• Hjorth "A Boundedness Lemma for Iterations"
• Zhu "Iterates of $M_1$"
Yizheng Zhu
Assume $\mathrm{AD}$. Let $\kappa$ be an inaccessible Suslin cardinal. Let $f \colon \kappa \to \kappa$ and let $\mu$ be a normal measure on $\kappa$ (e.g. $\mu$ can be any $\delta$-cofinality measure on $\kappa$). Let $\alpha = [f]_{\mu}$. Is there a characterization of the property $\alpha = w(\Delta)$, where $\Delta$ is a pointclass closed under $\exists^{\mathbb{R}}, \forall^{\mathbb{R}}$?
In particular, for $\mu =$ the $\omega$-cofinality measure on $\kappa$, can one find an $f \colon \kappa \to \kappa$ $f > \mathrm{id}$ such that $f(\beta) = w(\Delta)$ for all $\beta$ and $[f] = w(\Delta)$?
Motivation: If the answer is positive, then Steel's pointclass conjecture fails.
Steve Jackson

# Contact

Stefan Mesken
Einsteinstraße 62
48149 Münster
Germany
email: s (dot) mesken (at) wwu (dot) de
phone: +49-251-83-33767
orcid.org/0000-0002-1179-7049