Lattices of Elementary Substructures
Location: Hill 705
Date & time: Monday, 31 October 2016 at 5:00PM - 5:11PM
Athar Abdul-Quader , CUNY Graduate Center: Given any model of Peano Arithmetic, the collection of its elementary substructures forms a lattice under inclusion. The lattice problem for models of PA asks which lattices can be represented as substructure lattices of some model of PA. This question dates back to Gaifman's work on minimal types, which showed that the lattice 2 (the two element chain) can be represented as a substructure lattice. Since then, there have been many important contributions to this problem, including by Paris, Wilkie, Mills and Schmerl, though it remains open. The study of this question involves knowledge of models of PA as well as some nontrivial lattice theory and combinatorics. In this talk I will survey some of the major results and give a flavor of some of the techniques used in constructing models with prescribed substructure lattices. If there is time, I hope to describe how I have applied these techniques to construct a model of PA whose substructure lattice includes an infinite descending chain.