Transplendent Models: Expansions Omitting a Type

Abstract

We expand the notion of resplendency to theories of the kind $T+\mathrm{p\uparrow }$, where $T$ is a first-order theory and $\mathrm{p\uparrow }$ expresses that the type $p$ is omitted; both $T$ and $p$ are in languages extending the base language. We investigate two different formulations and prove necessary and sufficient conditions for countable recursively saturated models of PA.