Partially-elementary end extensions of countable admissible sets

TL;DR

This paper explores the transferability of properties from countable admissible sets to their partially-elementary end extensions, revealing limitations and conditions under which such extensions preserve certain theories.

Contribution

It demonstrates that some admissible sets lack certain elementary end extensions, but also shows conditions where such extensions do exist, clarifying the boundaries of transferability.

Findings

Some admissible sets with full separation lack certain elementary end extensions.

Existence of elementary end extensions depends on the theory and properties satisfied by the set.

Conditions for the existence of elementary end extensions are characterized in terms of $ ext{Pi}_n$-Collection and other properties.

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 $T$.