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 $\Pi_n\text{-}\mathsf{Collection}$ denote the restriction of the collection scheme to $\Pi _n$-formulae. We prove that for $n \geq 1$, if $\mathcal {M}$ is a model of $\mathsf{M}+\Pi _n\text{-}\mathsf{Collection}+{\mathsf{V}= \mathsf{L}}$ and $\mathcal {N}$ is a topless $\Sigma _{n+1}$-elementary end extension of $\mathcal {M}$ that satisfies $\Pi _{n-1}\text{-}\mathsf{Collection}$, then $\Pi _{n+1}\text{-}\mathsf{Collection}$ holds in $\mathcal {M}$. Here topless indicates that $\mathcal {N}$ contains an ordinal that is not in $\mathcal {M}$, but no least ordinal that is not in $\mathcal {M}$. This result is used to show that for $n \geq 1$, the minimum model of $\mathsf{M}+\Pi_n\text{-}\mathsf{Collection}$ has no $\Sigma _{n+1}$-elementary end extension that satisfies $\Pi _{n-1}\text{-}\mathsf{Collection}$, providing a negative answer to the generalisation of a question posed by Kaufmann.
