Notes on the equiconsistency of ZFC without the Power Set axiom and second order PA
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914086638321664 |
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| author | Kanovei, Vladimir Lyubetsky, Vassily |
| author_facet | Kanovei, Vladimir Lyubetsky, Vassily |
| contents | We demonstrate that theories $\text{Z}^-$, $\text{ZF}^-$, $\text{ZFC}^-$ (minus means the absence of the Power Set axiom) and $\text{PA}_2$, $\text{PA}_2^-$ (minus means the absence of the Countable Choice schema) are equiconsistent to each other. The methods used include the interpretation of a power-less set theory in $\text{PA}_2^-$ via well-founded trees, as well as the Gödel constructibility in the said power-less set theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_11643 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Notes on the equiconsistency of ZFC without the Power Set axiom and second order PA Kanovei, Vladimir Lyubetsky, Vassily Logic 03E25, 03E35, 03F35 (Primary), 03E15 (Secondary) We demonstrate that theories $\text{Z}^-$, $\text{ZF}^-$, $\text{ZFC}^-$ (minus means the absence of the Power Set axiom) and $\text{PA}_2$, $\text{PA}_2^-$ (minus means the absence of the Countable Choice schema) are equiconsistent to each other. The methods used include the interpretation of a power-less set theory in $\text{PA}_2^-$ via well-founded trees, as well as the Gödel constructibility in the said power-less set theory. |
| title | Notes on the equiconsistency of ZFC without the Power Set axiom and second order PA |
| topic | Logic 03E25, 03E35, 03F35 (Primary), 03E15 (Secondary) |
| url | https://arxiv.org/abs/2507.11643 |