# Research

I study the model theory of Peano Arithmetic (PA). Currently, I am focused on the connection between the “Lattice Problem for models of PA” and the question of which sets can be coded in elementary end extensions.

Given a model of PA, the collection of its elementary submodels forms a lattice under inclusion, called the “substructure lattice”. Given an elementary extension $\mathcal{M} \prec \mathcal{N}$ of models of PA, the collection of all models $\mathcal{K}$ such that $\mathcal{M} \preceq \mathcal{K} \preceq \mathcal{N}$ also forms a lattice, called the “interstructure lattice”, and is denoted Lt($\mathcal{N} / \mathcal{M}$).

Another invariant of an elementary (end) extension is the collection of subsets of the ground model “coded” in the extensions. To make this more precise, given $\mathcal{M} \prec_\text{end} \mathcal{N}$ the set $\{ X \cap M : X \in \textrm{Def}(\mathcal{N}) \}$ is denoted Cod($\mathcal{N} / \mathcal{M}$). It’s not clear, in general, if there is a connection between these two invariants. A recent paper by Jim Schmerl studies this connection for minimal end extensions (that is, end extensions realizing the interstructure lattice 2).