Partially-elementary end extensions of countable admissible sets

Abstract

A result of Kaufmann shows that if $L_α$ is countable, admissible and satisfies $Π_n\textsf{-Collection}$, then $\langle L_α, \in \rangle$ has a proper $Σ_{n+1}$-elementary end extension. This paper investigates to what extent the theory that holds in $\langle L_α, \in \rangle$ can be transferred to the partially-elementary end extensions guaranteed by Kaufmann's result. We show that there are $L_α$ satisfying full separation, powerset and $Π_n\textsf{-Collection}$ that have no proper $Σ_{n+1}$-elementary end extension satisfying either $Π_{n}\textsf{-Collection}$ or $Π_{n+3}\textsf{-Foundation}$. In contrast, we show that if $A$ is a countable admissible set that satisfies $Π_n\textsf{-Collection}$ and $T$ is a recursively enumerable theory that holds in $\langle A, \in \rangle$, then $\langle A, \in \rangle$ has a proper $Σ_n$-elementary end extension that satisfies $T$.