Partially-elementary end extensions of countable models of set theory

Let $\mathsf{K}\mathsf{P}$ denote Kripke-Platek Set Theory and let $\mathsf{M}$ be the weak set theory obtained from $\mathsf{Z}\mathsf{F}$ by removing the collection scheme, restricting separation to ${\mathrm{\Delta }}_0$-formulae and adding an axiom asserting that every set is contained in a transitive set ($\mathsf{T}\mathsf{C}\mathsf{o}$). A result due to Kaufmann shows that every countable model, $\mathcal{M}$, of $\mathsf{K}\mathsf{P}+{\mathrm{\Pi }}_n\mathsf{\text{-Collection}}$ has a proper ${\mathrm{\Sigma }}_{n+1}$-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 $\mathsf{K}\mathsf{P}+{\mathrm{\Pi }}_n\mathsf{\text{-Collection}}+{\mathrm{\Sigma }}_{n+1}\mathsf{\text{-Foundation}}$ and $T$ is a recursive theory that holds in $\mathcal{M}$, then there exists a proper ${\mathrm{\Sigma }}_n$-elementary end extension of $\mathcal{M}$ that satisfies $T$. We use this result to show that the theory $\mathsf{M}+{\mathrm{\Pi }}_n\mathsf{\text{-Collection}}+{\mathrm{\Pi }}_{n+1}\mathsf{\text{-Foundation}}$ proves ${\mathrm{\Sigma }}_{n+1}\mathsf{\text{-Separation}}$.