The set-theoretic Kaufmann-Clote question

TL;DR

This paper investigates the properties of certain set-theoretic models obtained by modifying ZF, focusing on how collection axioms behave under elementary end extensions, and provides a negative answer to a generalized Kaufmann question.

Contribution

It proves a new set-theoretic result about models of a weakened ZF and their extensions, extending understanding of collection axioms in set theory.

Findings

Models of weakened ZF with collection axioms have specific extension properties.

The minimum model of certain set theories cannot have particular elementary end extensions.

Provides a negative answer to a generalized Kaufmann question about set-theoretic models.

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