The set-theoretic Kaufmann-Clote question
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866916848880058368 |
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| author | McKenzie, Zachiri |
| author_facet | McKenzie, Zachiri |
| contents | Let $\mathsf{M}$ be the set theory obtained from $\mathsf{ZF}$ by removing the collection scheme, restricting separation to $Δ_0$-formulae and adding an axiom asserting that every set is contained in a transitive set. Let $Π_n\textsf{-Collection}$ denote the restriction of the collection scheme to $Π_n$-formulae. In this paper we prove that for $n \geq 1$, if $\mathcal{M}$ is a model of $\mathsf{M}+Π_n\textsf{-Collection}+\mathsf{V=L}$ and $\mathcal{N}$ is a $Σ_{n+1}$-elementary end extension of $\mathcal{M}$ that satisfies $Π_{n-1}\textsf{-Colelction}$ and that contains a new ordinal but no least new ordinal, then $Π_{n+1}\textsf{-Collection}$ holds in $\mathcal{M}$. This result is used to show that for $n \geq 1$, the minimum model of $\mathsf{M}+Π_n\textsf{-Collection}$ has no $Σ_{n+1}$-elementary end extension that satisfies $Π_{n-1}\textsf{-Collection}$, providing a negative answer to the generalisation of a question posed by Kaufmann. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_01176 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The set-theoretic Kaufmann-Clote question McKenzie, Zachiri Logic 03C62, 03H99 Let $\mathsf{M}$ be the set theory obtained from $\mathsf{ZF}$ by removing the collection scheme, restricting separation to $Δ_0$-formulae and adding an axiom asserting that every set is contained in a transitive set. Let $Π_n\textsf{-Collection}$ denote the restriction of the collection scheme to $Π_n$-formulae. In this paper we prove that for $n \geq 1$, if $\mathcal{M}$ is a model of $\mathsf{M}+Π_n\textsf{-Collection}+\mathsf{V=L}$ and $\mathcal{N}$ is a $Σ_{n+1}$-elementary end extension of $\mathcal{M}$ that satisfies $Π_{n-1}\textsf{-Colelction}$ and that contains a new ordinal but no least new ordinal, then $Π_{n+1}\textsf{-Collection}$ holds in $\mathcal{M}$. This result is used to show that for $n \geq 1$, the minimum model of $\mathsf{M}+Π_n\textsf{-Collection}$ has no $Σ_{n+1}$-elementary end extension that satisfies $Π_{n-1}\textsf{-Collection}$, providing a negative answer to the generalisation of a question posed by Kaufmann. |
| title | The set-theoretic Kaufmann-Clote question |
| topic | Logic 03C62, 03H99 |
| url | https://arxiv.org/abs/2507.01176 |