## Filippo Calderoni – The bi-embeddability relation for countable abelian groups

Talk held by Filippo Calderoni (Università di Torino, Italy and Politecnico di Torino, Italy) at the KGRC seminar on 2018-04-26.

Abstract: We analyze the Borel complexity of the bi‑embeddability relation for different classes of countable abelian groups. Most notably, we use the Ulm theory to prove that bi‑embeddability is incomparable with isomorphism in the case of p‑groups, and torsion groups. As I will explain, our result contrasts the arguable thesis that the bi‑embeddability relation on countable abelian p‑groups has strictly simpler complete invariants than isomorphism.

This is joint work with Simon Thomas.

## Diana Carolina Montoya: On some ideals associated with independent families

Talk held by Diana Carolina Montoya (KGRC) at the KGRC research seminar on 2018-04-19.

Title: On some ideals associated with independent families

Abstract. The concept of independence was first introduced by Fichtenholz and Kantorovic to study the space of linear functionals on the unit interval. Since then, independent families have been an important object of study in the combinatorics of the real line. Particular interest has been given, for instance, to the study of their definability properties and to their possible sizes.

In this talk we focus on two ideals which are naturally associated with independent families: The first of them is characterized by a diagonalization property, which allows us to add a maximal independent family along a finite support iteration of some ccc posets. The second ideal originates in Shelah’s proof of the consistency of $\mathfrak i\lt \mathfrak u$ (here $\mathfrak i$ and $\mathfrak u$ are  the independence and ultrafilter numbers respectively). Additionally, we study the relationship  between these two ideals for an arbitrary independent family $A$, and define a class of maximal  independent families — which we call densely independent — for which the ideals mentioned above  coincide. Building upon the techniques of Shelah we (1) characterize Sacks indestructibility for  such families in terms of properties of its associated diagonalization ideal, and (2) devise a countably closed poset which adjoins a Sacks indestructible densely maximal independent family.

This is joint work with Vera Fischer.

## Victoria Gitman – Virtual large cardinal principles

Research Seminar, Kurt Gödel Research Center, Thursday, April 12

Speaker: Victoria Gitman, (Graduate Center, City University of New York (CUNY), USA)

Abstract: Given a set-theoretic property $\mathcal P$ characterized by the existence of elementary embeddings between some first-order structures, we say that $\mathcal P$ holds virtually if the embeddings between structures from $V$ characterizing $\mathcal P$ exist somewhere in the generic multiverse. We showed with Schindler that virtual versions of supercompact, $C^{(n)}$-extendible, $n$-huge and rank-into-rank cardinals form a large cardinal hierarchy consistent with $V=L$. Sitting atop the hierarchy are virtual versions of inconsistent large cardinal principles such as the existence of an elementary embedding $j:V_\lambda\to V_\lambda$ for $\lambda$ much larger than the supremum of the critical sequence. The Silver indiscernibles, under $0^\sharp$, which have a number of large cardinal properties in $L$,are also natural examples of virtual large cardinals. With Bagaria, Hamkins and Schindler, we investigated properties of the virtual version of Vopenka’s Principle, which is consistent with $V=L$, and established some surprising differences from Vopenka’s Principle, stemming from the failure of Kunen’s Inconsistency in the virtual setting. A recent new direction in the study of virtual large cardinal principles involves asking that the required embeddings exist in forcing extensions preserving a large segment of the cardinals. In the talk, I will discuss a mixture of results about the virtual large cardinal hierarchy and virtual Vopenka’s Principle. Time permitting, I will give an overview of Woodin’s new results on virtual large cardinals in cardinal preserving extensions.we investigated properties of the virtual version of Vopenka’s Principle, which is consistent with $V=L$, and established some surprising differences from Vopenka’s Principle, stemming from the failure of Kunen’s Inconsistency in the virtual setting. A recent new direction in the study of virtual large cardinal principles involves asking that the required embeddings exist in forcing extensions preserving a large segment of the cardinals. In the talk, I will discuss a mixture of results about the virtual large cardinal hierarchy and virtual Vopenka’s Principle.

# Set theory today: A conference in honor of Georg Cantor

## Confirmed speakers:

• Omer Ben Neria, UCLA.
• Jörg Brendle, Kobe University.
• David Chodounský, Institute of Mathematics of the Czech Academy of Sciences.
• James Cummings, Carnegie Mellon University.
• Oswaldo Guzmán, York University.
• Radek Honzik, Charles University in Prague.
• Yurii Khomskii, University of Hamburg.
• Paul Larson, Miami University.
• Diego Mejía, Shizuoka University.
• Julien Melleray, University of Lyon.
• Heike Mildenberger, University of Freiburg.
• Luca Motto Ros, University of Torino.
• Christian Rosendal, University of Illinois at Chicago.
• Grigor Sargsyan, Rutgers University.
• Asger Törnquist, University of Copenhagen.
• Todor Tsankov, University of Paris Diderot.
• Matteo Viale, University of Torino.
• Jindřich Zapletal, University of Florida.

For a list of confirmed participants please see

## Organizing/scientific committee

• Vera Fischer
• Sy-David Friedman
• Benjamin Miller

## Local organizing committee

• Diana Carolina Montoya
• Daniel Soukup

## Monroe Eskew: Local saturation of the nonstationary ideals

Talk held by Monroe Eskew (KGRC) at the KGRC seminar on 2018-03-22.

Abstract: It is consistent relative to a huge cardinal that for all successor cardinals $\kappa$, there is a stationary $S \subseteq \kappa$ such that the nonstationary ideal on $\kappa$ restricted to $S$ is $\kappa^+$-saturated. We will describe the construction of the model, focusing how to get this property on all $\aleph_n$ simultaneously. Time permitting, we will also briefly discuss the Prikry-type forcing that extends this up to $\aleph_{\omega+1}$.

## Thilo Weinert: Cardinal Characteristics and Partition Properties

Talk held by Thilo Weinert (KGRC) at the KGRC seminar on 2018-03-15.

Abstract: Many a partition relation has been proved assuming the Generalised Continuum Hypothesis. More precisely, many negative partition relations involving ordinals smaller than $\omega_2$ have been proved assuming the Continuum Hypothesis. Some recent results in this vein for polarised partition relations came from Garti and Shelah. The talk will focus on classical partition relations. The relations $\omega_1\omega \not\rightarrow (\omega_1\omega, 3)^2$ and $\omega_1^2 \not\rightarrow (\omega_1\omega, 4)^2$ were both shown to follow from the Continuum Hypothesis, the former in 1971 by Erdős and Hajnal and the latter in 1987 by Baumgartner and Hajnal.

The former relation was shown to follow from both the dominating number and the stick number being $\aleph_1$ in 1987 by Takahashi. In 1998 Jean Larson showed that simply the dominating number being $\aleph_1$ suffices for this. It turns out that the unbounding number and the stick number both being $\aleph_1$ yields the same result. Moreover, also the second relation follows both from the dominating number being  $\aleph_1$ and from both the unbounding number and the stick number being $\aleph_1$ thus answering a question of Jean Larson.

This is both joint work with Chris Lambie-Hanson and with both William Chen and Shimon Garti.

## Šárka Stejskalová – The tree property and the continuum function

Talk held by Šárka Stejskalová (KGRC) at the KGRC seminar on 2018-03-08.

Abstract: We will discuss the tree property, a compactness principle which can hold at successor cardinals such as $\aleph_2$ or $\aleph_3$. For a regular cardinal $\kappa$, we say that $\kappa$ has the tree property if there are no $\kappa$-Aronszajn trees. It is known that the tree property has the following non-trivial effect on
the continuum function:

(*) If the tree property holds at $\kappa^{++}$, then $2^\kappa> \kappa^+$.

After defining the key notions, we will review some basic constructions related to the tree property and state some original results regarding the tree property which suggest that (*) is the only restriction which the tree property puts on the continuum function in addition to the usual restrictions provable in ZFC.

## David Schrittesser – On the Complexity of Maximal Cofinitary Groups

Talk held by David Schrittesser (KGRC) at the KGRC seminar on 2018-01-11.

Abstract: A maximal cofinitary group is a subgroup of the group of permutations of the set of natural numbers $\mathbb N$ such that any group element has only finitely many fixed points, and no strictly larger group of permutations of $\mathbb N$ has this property. Improving a result of Horowitz and Shelah, we show that there is a closed maximal cofinitary group.

## Moritz Müller: On the relative strength of finitary combinatorial principles

Talk held by Moritz Müller (KGRC) at the KGRC seminar on 2017-12-14 at 4pm .

Abstract: Define a finitary combinatorial principle to be a first-order sentence which is valid in the finite but falsifiable in the infinite. We aim to compare the strength of such principles over a weak arithmetic. We distinguish “weak” and “strong” principles based on their behaviour with respect to finite structures that are only partially defined. The talk sketches a forcing proof of a theorem stating that over relativized $T^1_2$ “weak” principles do not imply “strong” ones.

## Maxwell Levine: Forcing Square Sequences

KGRC research seminar on 2017-11-30 at 4pm.

Speaker: Maxwell Levine (KGRC)

Abstract: In the 1970’s, Jensen proved that Gödel’s constructible universe $L$ satisfies a combinatorial principle called $\square_\kappa$ for every uncountable cardinal $\kappa$. Its significance is partially in that it clashes with the reflection properties of large cardinals – for example, if $\mu$ is supercompact and $\kappa \ge \mu$ then $\square_\kappa$ fails – and so it characterizes the minimality of $L$ in an indirect way. Schimmerling devised an intermediate hierarchy of principles $\square_{\kappa,\lambda}$ for $\lambda \le \kappa$ as a means of comparing a given model of set theory to $L$, the idea being that a smaller value of $\lambda$ yields a model that is more similar to $L$ at $\kappa$.

Cummings, Foreman, and Magidor proved that for any $\lambda<\kappa$, $\square_{\kappa,\lambda}$ implies the existence of a PCF-theoretic object called a very good scale for $\kappa$, but that $\square_{\kappa,\kappa}$ (usually denoted $\square_\kappa^\ast$) does not. They asked whether $\square_{\kappa,<\kappa}$ implies the existence of a very good scale for $\kappa$, and we resolve this question in the negative.

We will discuss the technical background of the problem, provide a complete solution, and discuss further avenues of research.