Partially-elementary end extensions of countable models of set theory

Abstract

Let KP denote Kripke–Platek Set Theory and let M be the weak set theory obtained from 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 [9] shows that every countable model, M of KP+\Pi _n\textsf {-Collection}$ has a proper $\Sigma _{n+1}$ -elementary end extension. We show that for all $n \geq 1$, there exists an $L_\alpha $ (where $L_\alpha $ is the $\alpha ^{\textrm {th}}$ approximation of the constructible universe L ) that satisfies Separation, Powerset and $\Pi _n\textsf {-Collection}$, but that has no $\Sigma _{n+1}$ -elementary end extension satisfying either $\Pi _n\textsf {-Collection}$ or $\Pi _{n+3}\textsf {-Foundation}$. Thus showing 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 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 $\Sigma _{n+1}\textsf {-Separation}$.