The set-theoretic Kaufmann–Clote question

Abstract

Let M be the set theory obtained from ZF by removing the collection scheme, restricting separation to Δ0-formulae and adding an axiom asserting that every set is contained in a transitive set. Let Πn-Collection denote the restriction of the collection scheme to Πn-formulae. We prove that for n≥1, if M is a model of M+Πn-Collection+V=L and N is a topless Σn+1-elementary end extension of M that satisfies Πn−1-Collection, then Πn+1-Collection holds in M. Here topless indicates that N contains an ordinal that is not in M, but no least ordinal that is not in M. This result is used to show that for n≥1, the minimum model of M+Πn-Collection has no Σn+1-elementary end extension that satisfies Πn−1-Collection, providing a negative answer to the generalisation of a question posed by Kaufmann.