Partially-elementary end extensions of countable models of set theory

Zachiri McKenzie

Published 2024-06-26Version 1

Let $\mathsf{KP}$ denote Kripke-Platek Set Theory and let $\mathsf{M}$ be the weak set theory obtained from $\mathsf{ZF}$ by removing the collection scheme, restricting separation to $\Delta_0$-formulae and adding an axiom asserting that every set is contained in a transitive set ($\mathsf{TCo}$). A result due to Kaufmann shows that every countable model, $\mathcal{M}$, of $\mathsf{KP}+\Pi_n\textsf{-Collection}$ has a proper $\Sigma_{n+1}$-elementary end extension. Here we show that there are limits to the amount of the theory of $\mathcal{M}$ that can be transferred to the end extensions that are guaranteed by Kaufmann's Theorem. Using admissible covers and the Barwise Compactness Theorem, we show that if $\mathcal{M}$ is a countable model $\mathsf{KP}+\Pi_n\textsf{-Collection}+\Sigma_{n+1}\textsf{-Foundation}$ and $T$ is a recursive theory that holds in $\mathcal{M}$, then there exists a proper $\Sigma_n$-elementary end extension of $\mathcal{M}$ that satisfies $T$. We use this result to show that the theory $\mathsf{M}+\Pi_n\textsf{-Collection}+\Pi_{n+1}\textsf{-Foundation}$ proves $\Sigma_{n+1}\textsf{-Separation}$.