On the categoricity of complete second order theories
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866911868393619456 |
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| author | Saarinen, Tapio Väänänen, Jouko Woodin, William Hugh |
| author_facet | Saarinen, Tapio Väänänen, Jouko Woodin, William Hugh |
| contents | We show, assuming PD, that every complete finitely axiomatized second order theory with a countable model is categorical, but that there is, assuming again PD, a complete recursively axiomatized second order theory with a countable model which is non-categorical. We show that the existence of even very large (e.g. supercompact) cardinals does not imply the categoricity of all finite complete second order theories. More exactly, we show that a non-categorical complete finitely axiomatized second order theory can always be obtained by (set) forcing. We also show that the categoricity of all finite complete second order theories with a model of a certain singular cardinality kappa of uncountable cofinality can be forced over any model of set theory. Previously, Solovay had proved, assuming V=L, that every complete finitely axiomatized second order theory (with or without a countable model) is categorical, and that in a generic extension of L there is a complete finitely axiomatized second order theory with a countable model which is non-categorical. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_03428 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the categoricity of complete second order theories Saarinen, Tapio Väänänen, Jouko Woodin, William Hugh Logic 03E35, 03E47, 03C85 We show, assuming PD, that every complete finitely axiomatized second order theory with a countable model is categorical, but that there is, assuming again PD, a complete recursively axiomatized second order theory with a countable model which is non-categorical. We show that the existence of even very large (e.g. supercompact) cardinals does not imply the categoricity of all finite complete second order theories. More exactly, we show that a non-categorical complete finitely axiomatized second order theory can always be obtained by (set) forcing. We also show that the categoricity of all finite complete second order theories with a model of a certain singular cardinality kappa of uncountable cofinality can be forced over any model of set theory. Previously, Solovay had proved, assuming V=L, that every complete finitely axiomatized second order theory (with or without a countable model) is categorical, and that in a generic extension of L there is a complete finitely axiomatized second order theory with a countable model which is non-categorical. |
| title | On the categoricity of complete second order theories |
| topic | Logic 03E35, 03E47, 03C85 |
| url | https://arxiv.org/abs/2405.03428 |