The set-theoretic Kaufmann-Clote question

Abstract

Let $\mathsf{M}$ be the 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. Let ${\Pi }_n\text{-Collection}$ denote the restriction of the collection scheme to ${\Pi }_n$-formulae. In this paper we prove that for $n\ge 1$, if $\mathcal{M}$ is a model of $\mathsf{M}+{\Pi }_n\text{-Collection}+\mathsf{V}=\mathsf{L}$ and $\mathcal{N}$ is a ${\Sigma }_{n+1}$-elementary end extension of $\mathcal{M}$ that satisfies ${\Pi }_{n-1}\text{-Colelction}$ and that contains a new ordinal but no least new ordinal, then ${\Pi }_{n+1}\text{-Collection}$ holds in $\mathcal{M}$. This result is used to show that for $n\ge 1$, the minimum model of $\mathsf{M}+{\Pi }_n\text{-Collection}$ has no ${\Sigma }_{n+1}$-elementary end extension that satisfies ${\Pi }_{n-1}\text{-Collection}$, providing a negative answer to the generalisation of a question posed by Kaufmann.

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