## 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.

## Matt Foreman: Games on weakly compact cardinals

TAU Forcing Seminar

Tuesday, 25/12/18

Speaker: Matt Foreman

Title: Games on weakly compact cardinals

Abstract: Attached.

## Assaf Rinot: Hindman’s theorem and uncountable Abelian groups

Colloquium, Hebrew University of Jerusalem

Thu, 20/12/2018 – 14:30 to 15:30

Location:
Manchester Building (Hall 2), Hebrew University Jerusalem

Speaker: Assaf Rinot

Title: Hindman’s theorem and uncountable Abelian groups

Abstract. In the early 1970’s, Hindman proved a beautiful theorem in additive Ramsey theory asserting that for any partition of the set of natural numbers into finitely many cells, there exists some infinite set such that all of its finite sums belong to a single cell. In this talk, we shall address generalizations of this statement to the realm of the uncountable. Among other things, we shall present a negative partition relation for the real line which simultaneously generalizes a recent theorem of Hindman, Leader and Strauss, and a classic theorem of Galvin and Shelah. This is joint work with David Fernandez-Breton.

## Asaf Karagila: On countable unions of countable sets

BIU seminar in Set Theory

December 17, 2018

Speaker: Asaf Karagila (UEA)

Title: On countable unions of countable sets

Abstract. How big can countable unions of countable sets be? Assuming the axiom of choice, countable. Not assuming the axiom of choice, it is not hard to arrange situation where there are many incomparable cardinals which are the countable union of countable sets. But none of them are “particularly large”. While a countable union of countable sets can at most be mapped onto $\omega_1$, its power set can be made much larger. We prove an old (and nearly forgotten) theorem of Douglass Morris, that it is consistent that for every $\alpha$ there is a set which is a countable union of countable sets, but its power set can be mapped onto $\alpha$.

## Ilijas Farah: On the model theory of C*-algebras

HUJI Logic Seminar
12/December/18, 11 am, in Ross 63.

Speaker: Ilijas Farah

Title: On the model theory of C*-algebras

Abstract. Ultrapowers and reduced products play a central role in the Elliott classification program for separable (nuclear, etc.) C*-algebras. Although an ultrapower of a separable C*-algebra A is quite different from the reduced product ℓ∞(A)/c0(A)

, these massive algebras are interchangeable in many (but not quite all) concrete applications. I will present a theorem
that attempts to give an abstract explanation of this phenomenon. This preliminary result applies to some other axiomatizable categories, and its proof does not use any of the nontrivial theory of C*-algebras.

No previous knowledge of C*-algebras is required; they appear primarily as a motivation.
This is preliminary part of a joint work with Christopher Schafhauser.

## Saharon Shelah: The spectrum of the existence of a universal model

HUJI Logic Seminar

21 November 2018

Speaker: Saharon Shelah

Title: The spectrum of the existence of a universal model

Abstract. The existence of a universal model (of a theory T in a cardinal lambda) is a natural question in model theory and set theory. We shall deal with new sufficient conditions for non-existence.

## Yair Hayut: Chang’s conjecture

HUJI Set Theory Seminar

November 21, 2018

Speaker: Yair Hayut

Abstract. I will review some consistency results related to Chang’s Conjecture (CC). First I will discuss some classical results of deriving instances of CC from huge cardinals and the new results for getting instances of CC from supercompact cardinals, and present some open problems. Then, I will review the consistency proof of some versions of the Global Chang’s Conjecture – which is the consistency of the occurrence many instances of CC simultaneously. We will aim to show the consistency of the statement: (\mu^+,\mu) –>> (\nu^+,\nu) for all regular \mu and all \nu < \mu, starting from a huge cardinal. In order to prove this we will start with the easier task in which $\mu$ is assumed to be regular. In order to get the stronger result, we will force with Radin forcing over a model in which many instances of CC hold.

## Shimon Garti: Cardinal invariants for singular cardinals

HUJI Set Theory Seminar

November 14, 2018

Speaker: Shimon Garti

Abstract. We shall try to prove the consistency of d_lambda > r_lambda (and even d_lambda > u_lambda) for a singular cardinal lambda.
This is a joint work with Saharon.