MOPA

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.

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.

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.

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.

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

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.