A note on Łoś's Theorem without the Axiom of Choice
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866913463614308352 |
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| author | Usuba, Toshimichi |
| author_facet | Usuba, Toshimichi |
| contents | We study some topics about Łoś's theorem without assuming the Axiom of Choice. We prove that Łoś's fundamental theorem of ultraproducts is equivalent to a weak form that every ultrapower is elementary equivalent to its source structure. On the other hand, it is consistent that there is a structure $M$ and an ultrafilter $U$ such that the ultrapower of $M$ by $U$ is elementary equivalent to $M$, but the fundamental theorem for the ultrapower of $M$ by $U$ fails. We also show that weak fragments of the Axiom of Choice, such as the Countable Choice, do not follow from Łoś's theorem, even assuming the existence of non-principal ultrafilters. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_14267 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A note on Łoś's Theorem without the Axiom of Choice Usuba, Toshimichi Logic 03C20, 03E25, 03E35, 03E55 We study some topics about Łoś's theorem without assuming the Axiom of Choice. We prove that Łoś's fundamental theorem of ultraproducts is equivalent to a weak form that every ultrapower is elementary equivalent to its source structure. On the other hand, it is consistent that there is a structure $M$ and an ultrafilter $U$ such that the ultrapower of $M$ by $U$ is elementary equivalent to $M$, but the fundamental theorem for the ultrapower of $M$ by $U$ fails. We also show that weak fragments of the Axiom of Choice, such as the Countable Choice, do not follow from Łoś's theorem, even assuming the existence of non-principal ultrafilters. |
| title | A note on Łoś's Theorem without the Axiom of Choice |
| topic | Logic 03C20, 03E25, 03E35, 03E55 |
| url | https://arxiv.org/abs/2311.14267 |