December 4
Athar Abdul-Quader,
Purchase College
The pentagon lattice
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.