On Cofinal Submodels and Elementary Interstices

Abstract

We prove a number of results concerning the variety of first-order theories and isomorphism types of pairs of the form $(N,M)$, where $N$ is a countable recursively saturated model of Peano Arithmetic and $M$ is its cofinal submodel. We identify two new isomorphism invariants for such pairs. In the strongest result we obtain continuum many theories of such pairs with the fixed greatest common initial segment of $N$ and $M$ and fixed lattice of interstructures $K$, such that $M\prec K\prec N$.