# The strength of Sealing

Nam Trang

## Institution:

University of North Texas, Denton

## Time:

Monday, May 20, 2019 - 4:00pm to 5:30pm

## Location:

RH440R

Sealing is a type of generic absoluteness condition introduced by Woodin that asserts in strong terms that the theory of the universally Baire sets cannot be changed by forcing. generic-LSA is the statement that the Largest Suslin Axiom (LSA) holds in all generic extensions. Under a mild large cardinal hypothesis, we show that Sealing is equiconsistent with generic-LSA. As a consequence, Sealing is weaker than the theory ZFC+there is a Woodin cardinal which is a limit of Woodin cardinals". Sealing's consistency being weak represents an obstruction to the current program of descriptive inner model theory. Going beyond this bound in core model induction applications seems challenging and requires us to construct third order objects (subsets of the universally Baire sets). We will state the precise theorems and discuss their impact on current developments of inner model theory. Time allowed, we will talk a bit about how to overcome the the obstructions imposed by Sealing. This is joint work with G. Sargsyan.

# TBA

Kevin Duanmu

UC Berkeley

## Time:

Monday, October 14, 2019 - 4:00pm

TBA

# Precipitous Ideals III

## Speaker:

Matthew D. Foreman

UCI

## Time:

Monday, March 11, 2019 - 4:00pm to 5:30pm

## Location:

RH 440R

This lecture will complete the presentation of a new result answering a question of Welch about precipitous ideals on weakly compact cardinals.

# Precipitous Ideals II

## Speaker:

Matthew D. Foreman

UCI

## Time:

Monday, March 4, 2019 - 4:00pm to 5:30pm

## Location:

RH 440R

This lecture will present a new result answering a question of Welch about precipitous ideals on weakly compact cardinals.

# Precipitous Ideals I

## Speaker:

Matthew D. Foreman

UCI

## Time:

Monday, February 25, 2019 - 4:00pm to 5:30pm

## Location:

RH 440R

This lecture will review the definitions and basic facts about precipitous ideals and generic elementary embeddings.

# Omitting types in the logic of metric structures (Joint work with I. Farah)

Menachem Magidor

## Institution:

Einstein Institute of Mathematics

## Time:

Monday, February 4, 2019 - 4:00pm to 5:00pm

## Location:

440R RH

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.

# Strongly minimal sets in continuous logic

James Hanson

## Institution:

University of Wisconsin

## Time:

Monday, April 8, 2019 - 4:00pm to 5:00pm

## Location:

RH 440R

The precise structural understanding of uncountably categorical theories given by the Baldwin-Lachlan theorem is known to fail in continuous logic in the context of inseparably categorical theories. The primary obstacle is the absence of strongly minimal sets in some inseparably categorical theories. We will develop the concept of strongly minimal sets in continuous logic and discuss some common conditions under which they are present in an ω-stable metric theory. Finally we will examine the extent to which we recover the Baldwin-Lachlan theorem in the presence of strongly minimal sets.