Partially-elementary end extensions of countable models of set theory
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914009204129792 |
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| author | McKenzie, Zachiri |
| author_facet | McKenzie, Zachiri |
| contents | Let $\mathsf{KP}$ denote Kripke-Platek Set Theory and let $\mathsf{M}$ be the weak 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 ($\mathsf{TCo}$). A result due to Kaufmann shows that every countable model, $\mathcal{M}$, of $\mathsf{KP}+Π_n\textsf{-Collection}$ has a proper $Σ_{n+1}$-elementary end extension. Here we show that there are limits to the amount of the theory of $\mathcal{M}$ that can be transferred to the end extensions that are guaranteed by Kaufmann's Theorem. Using admissible covers and the Barwise Compactness Theorem, we show that if $\mathcal{M}$ is a countable model $\mathsf{KP}+Π_n\textsf{-Collection}+Σ_{n+1}\textsf{-Foundation}$ and $T$ is a recursive theory that holds in $\mathcal{M}$, then there exists a proper $Σ_n$-elementary end extension of $\mathcal{M}$ that satisfies $T$. We use this result to show that the theory $\mathsf{M}+Π_n\textsf{-Collection}+Π_{n+1}\textsf{-Foundation}$ proves $Σ_{n+1}\textsf{-Separation}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_18341 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Partially-elementary end extensions of countable models of set theory McKenzie, Zachiri Logic 03C62, 03H99, 03C70 Let $\mathsf{KP}$ denote Kripke-Platek Set Theory and let $\mathsf{M}$ be the weak 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 ($\mathsf{TCo}$). A result due to Kaufmann shows that every countable model, $\mathcal{M}$, of $\mathsf{KP}+Π_n\textsf{-Collection}$ has a proper $Σ_{n+1}$-elementary end extension. Here we show that there are limits to the amount of the theory of $\mathcal{M}$ that can be transferred to the end extensions that are guaranteed by Kaufmann's Theorem. Using admissible covers and the Barwise Compactness Theorem, we show that if $\mathcal{M}$ is a countable model $\mathsf{KP}+Π_n\textsf{-Collection}+Σ_{n+1}\textsf{-Foundation}$ and $T$ is a recursive theory that holds in $\mathcal{M}$, then there exists a proper $Σ_n$-elementary end extension of $\mathcal{M}$ that satisfies $T$. We use this result to show that the theory $\mathsf{M}+Π_n\textsf{-Collection}+Π_{n+1}\textsf{-Foundation}$ proves $Σ_{n+1}\textsf{-Separation}$. |
| title | Partially-elementary end extensions of countable models of set theory |
| topic | Logic 03C62, 03H99, 03C70 |
| url | https://arxiv.org/abs/2406.18341 |