Partially-elementary end extensions of countable admissible sets

Abstract

A result of Kaufmann shows that if $L_{\alpha }$ is countable, admissible and satisfies ${\Pi }_n\text{-Collection}$, then $⟨L_{\alpha },\in ⟩$ has a proper ${\Sigma }_{n+1}$-elementary end extension. This paper investigates to what extent the theory that holds in $⟨L_{\alpha },\in ⟩$ 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 ${\Pi }_n\text{-Collection}$ that have no proper ${\Sigma }_{n+1}$-elementary end extension satisfying either ${\Pi }_n\text{-Collection}$ or ${\Pi }_{n+3}\text{-Foundation}$. In contrast, we show that if $A$ is a countable admissible set that satisfies ${\Pi }_n\text{-Collection}$ and $T$ is a recursively enumerable theory that holds in $⟨A,\in ⟩$, then $⟨A,\in ⟩$ has a proper ${\Sigma }_n$-elementary end extension that satisfies $T$.

13 pages