**CUNY Graduate Center**

**Room 4214.03**

**Wednesdays 6:30pm-8pm**

**Organized by Roman Kossak**

**Fall 2019**

**December 4**

**Athar Abdul-Quader**
Purchase College

**The pentagon lattice**

**Abstract**

Wilkie (1977) proved that if $M$ is a countable model of ${\rm PA}$, it has an elementary end extension $N$ such that the interstructure lattice $Lt(N / M)$ is the pentagon lattice $\mathbf{N}_5$. A similar result can be shown for cofinal extensions. Surprisingly, a new result by Jim Schmerl states that no model of ${\rm PA}$ has a 'mixed' elementary extension (one that is neither end nor cofinal) whose interstructure lattice is the pentagon. In this talk, I will go over the definitions and describe the method used in proofs about interstructure lattices.

**November 6**

**Roman Kossak**
CUNY

**Short recursively saturated models of PA as an AEC**

**Abstract**

Countable short recursively saturated models of PA can serve as bases for abstract elementary classes that are complete but not irreducible. I will explain all these notions and show the construction.

**October 16**

**Alf Dolich**
CUNY

**Getting Atomic Models of Size Continuum**

**Abstract**

Following Baldwin and Laskowski's 'Henkin Constructions of Models of Size Continuum' I will outline how the main results of this paper can be used to show that under a variety of assumptions a theory T with an uncountable atomic model also has an atomic model of size continuum.

**October 2**

**Corey Switzer**
CUNY

**Constructions of Size Continuum**

**Abstract**

I will give an exposition of a technique by Baldwin and Laskowski for extending the Henkin construction to get models of size continuum with interesting properties. The main theorem gives sufficient conditions for a theory to have a model of size continuum which is Borel, atomic and omits some given collection of countably many types. Time permitting I will sketch some applications as well.

**September 25**

**Alf Dolich**
CUNY

**Henkin Constructions of Models with Size Continuum after Baldwin and Laskowski II**

**Abstract**

I will continue last weeks talks addressing issues around the absoluteness of categoricity for sentences of infinitary logic.

**September 18**

**Alf Dolich**
CUNY

**Henkin Constructions of Models with Size Continuum after Baldwin and Laskowski**

**Abstract**

In recent work Baldwin and Laskowski introduced a method to construct via a Henkin-style construction models of size continuum in countable many steps. This construction has multiple applications. In this talk I will survey, following a lecture of Baldwin, some of the background and motivations for this construction.