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 of models of PA, the collection of all models such that also forms a lattice, called the “interstructure lattice”, and is denoted Lt().
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 the set is denoted Cod(). 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).