## Spencer Unger: Stationary reflection and the singular cardinals hypothesis

HUJI Logic Seminar

Tomorrow, Spencer Unger will speak in our logic seminar about Stationary reflection and the singular cardinals hypothesis. As usual we meet at 11am in Ross 63. Looking forward to seeing you there,

Title: Stationary reflection and the singular cardinals hypothesis.

Abstact. We examine reflection of stationary sets at successors of singular cardinals and its connection with cardinal arithmetic. For instance it has been open whether the failure of the singular cardinal hypothesis at a singular cardinal mu of uncountable cofinality implies the existence of a nonreflecting stationary subset of mu^+. In recent joint work with Omer Ben-Neria and Yair Hayut we have shown that the answer is no modulo the consistency of some large cardinals. In this talk, we survey some instances of methods used in the proof. In particular, we show how to construct Prikry sequences over iterated ultrapowers and exploit them for combinatorial proofs.

## Ralf Schindler: Variants of the extender algebra and their applications

BIU Infinite Combinatorics Seminar

Mon, 25/03/2019 – 13:00

Speaker: Ralf Schindler (Münster)

Title: Variants of the extender algebra and their applications

Abstract. In the 1970’ies, Bukowský identified a beautiful and handy criterion for when V is a forcing extension of a given inner model, which proved very useful recently in set theoretical geology. In the 1990’ies, Woodin isolated his extender algebra which makes use of a large cardinal, a Woodin cardinal. It turns out that Bukowský’s theorem and Woodin’s extender algebra may be presented in a uniform fashion – one proof and one forcing gives both results. We will present the proof and then discuss its application in inner model theoretic geology. This is joint work with Grigor Sargsyan and Farmer Schlutzenberg.

## Miguel Moreno: The Main Gap in the generalized Borel-reducibility hierarchy

BIU Infinite Combinatorics Seminar

Mon, 11/03/2019 – 13:00

Speaker: Miguel Moreno (BIU)

Title: The Main Gap in the generalized Borel-reducibility hierarchy

Abstract. During this talk we will discuss where in the generalized Borel-reducibility hierarchy are the isomorphism relation of first order complete theories. These theories are divided in two kind:classifiable and non-classifiable. To study the classifiable theories case is needed the use of Ehrenfeucht-Fraïssé games. On the other hand the study of the non-classifiable theories is done by using colored trees. The goal of the talk is to see the classifiable theories case and start the non-classifiable theories case by proving that it is possible to map every element of the generalized Baire, f, into a colored tree, J(f), such that; for every f and g elements of the generalized Baire space, J(f) and J(g) are isomorphic as colored trees if and only if f and g coincide on a club.

## Sandra Müller: Projective determinacy for games of length $\omega^2$ and longer

BIU Infinite Combinatorics Seminar

Sandra Müller (KGRC)
25/02/2019 – 13:0015:00

We will study infinite two player games and the large cardinal strength corresponding to their determinacy. For games of length $\omega$ this is well understood and there is a tight connection between the determinacy of projective games and the existence of canonical inner models with Woodin cardinals. For games of arbitrary countable length, Itay Neeman proved the determinacy of analytic games of length $\omega \cdot \theta$ for countable $\theta\> \omega$ from a sharp for $\theta$ Woodin cardinals.

We aim for a converse at successor ordinals. In joint work with Juan P. Aguilera we showed that determinacy of $\boldsymbol\Pi^1\_{n+1}$ games of length $\omega^2$ implies the existence of a premouse with $\omega+n$ Woodin cardinals. This generalizes to a premouse with $\omega+\omega$ Woodin cardinals from the determinacy of games of length $\omega^2$ with $\Game^{\mathbb{R}}\boldsymbol\Pi^1\_1$ payoff.

If time allows, we will also sketch how these methods can be adapted to, in combination with results of Nam Trang, obtain $\omega^\alpha+n$ Woodin cardinals for countable ordinals $\alpha$ and natural numbers $n$ from the determinacy of sufficiently long projective games.

## Dominik Adolf: The strength of very small Jonsson cardinals

BIU Infinite Combinatorics Seminar

Dominik Adolf (BIU)
18/02/2019 – 13:0015:00

An uncountable cardinal κ is Jonnson if only if the set of proper subsets of κ that are of cardinality κ is stationary. Though this property has large cardinal strength it is not at all clear that Jonnson cardinals do in fact need to be large in the obvious sense. For example, it is known that Jonsson cardinals can be singular.

In this talk we will use the methods of Inner Model Theory to show that, given the assumption that the least singular cardinal is Jonsson, there is a canonical model with a strong cardinal together with a class of Silver indiscernibles for this model. (The proof presented will make some simplifying assumptions.) Time permitting, we may discuss approaches to extend this result to show the existence of inner models with Woodin cardinals and more.

## Daniel T. Soukup: Uniformization properties and graph edge colourings

BIU Infinite Combinatorics Seminar

Daniel T. Soukup (KGRC)
11/02/2019 – 13:0015:00

Sierpinski’s now classical result states that there is an edge 2-colouring of the complete graph on aleph1 vertices so that there are no uncountable monochromatic subgraphs. In the 1970s, Erdos, Galvin and Hajnal asked what other graphs with large chromatic number admit similar edge colourings i.e., with no ‘large’ monochromatic subgraphs. We plan to review some recent advances on this problem and in particular, connect the question to Shelah’s ladder system uniformization theory.

## Piotr Szewczak: Selection Principles in Mathematics (an overview)

BIU Infinite Combinatorics Seminar

Piotr Szewczak, Cardinal Stefan Wyszyński University in Warsaw, Poland
04/02/2019 – 13:0015:00

The theory of selection principles deals with the possibility of obtaining mathematically significant objects by selecting elements from sequences of sets. The studied properties mainly include covering properties, measure- and category-theoretic properties, and local properties in topological spaces, especially functions spaces. Often, the characterization of a mathematical property using a selection principle is a nontrivial task leading to new insights on the characterized property.
I will give an overview of this theory and, if time permits, I will present some results obtained jointly with Boaz Tsaban and Lyubomyr Zdomskyy.

## Menachem Magidor: Omitting types in the logic of metric structures

HUJI Logic Seminar

16/Jan/2019, 11-13, Ross 63.

Speaker: Menachem Magidor

Title: Omitting types in the logic of metric structures

Abstract.

(joint work with I. Farah)

The logic of metric structures was introduced by Ben Yaacov, Berenstein , Henson and Usvyatsov. It is a version of continuous logic which allows fruitful model theory for many kinds of metric structures. There are many aspects of this logic which make it similar to first order logic, like compactness, a complete proof system, an omitting types theorem for complete types etc. But when one tries to generalize the omitting type criteria to general (non-complete) types the problem turns out to be essentially more difficult than the first order situation. For instance one can have two types (in a complete theory) that each one can be omitted, but they can not be omitted simultaneously.

In the beginning of the talk we shall give a brief survey of the logic of metric structures, so the talk should be accessible also the listeners who are not familiar with the logic of metric structures.

## Miguel Moreno: An introduction to generalized descriptive set theory, part 2

BIU Infinite Combinatorics Seminar

Date : 31/12/2018 – 13:00 – 15:00

Speaker: Miguel Moreno (BIU)

Title : An introduction to generalized descriptive set theory, part 2

Abstract. After introducing the notions of $\kappa$-Borel class, $\kappa$-$\Delta_1^1$ class, $\kappa$-Borel^* class in the previous talk, in this talk, we will show the relation between this classes.
In descriptive set theory the Borel class, the $\Delta_1^1$ class, the Borel* class are the same class, this doesn’t hold in the generalized descriptive set theory, in particular under the assumption V=L the Borel* class is equal to the $\Sigma1^ 1$ class.

## Ur Yaar: The Modal Logic of Forcing

HUJI Set Theory Seminar
On Wednesday, December 26, Ur Yaar will talk about the modal logic of forcing.

Title: The Modal Logic of Forcing

Abstract: Modal logic is used to study various modalities, i.e. various ways in which statements can be true, the most notable of which are the modalities of necessity and possibility. In set-theory, a natural interpretation is to consider a statement as necessary if it holds in any forcing extension of the world, and possible if it holds in some forcing extension. One can now ask what are the modal principles which captures this interpretation, or in other words – what is the “Modal Logic of Forcing”?
We can also restrict ourselves only to a certain class of forcing notions, or to forcing over a specific universe, resulting in an abundance of questions to be resolved.
We will begin with a short introduction to modal logic, and then present the tools developed by Joel Hamkins and Benedikt Loewe to answer these questions. We will present their answer to the original question, and then move to focus on the class of sigma-centered forcings, which I investigated in my Master’s thesis.