Partially-elementary end extensions of countable admissible sets

Abstract

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