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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2211.03976 |
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| _version_ | 1866909456578641920 |
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| author | Harrison-Trainor, Matthew Kulshreshtha, Dhruv |
| author_facet | Harrison-Trainor, Matthew Kulshreshtha, Dhruv |
| contents | We work in the setting of Zermelo-Fraenkel set theory without assuming the Axiom of Choice. We consider sets with the Boolean operations together with the additional structure of comparing cardinality (in the Cantorian sense of injections). What principles does one need to add to the laws of Boolean algebra to reason not only about intersection, union, and complementation of sets, but also about the relative size of sets? We give a complete axiomatization.
A particularly interesting case is when one restricts to the Dedekind-finite sets. In this case, one needs exactly the same principles as for reasoning about imprecise probability comparisons, the central principle being Generalized Finite Cancellation (which includes, as a special case, division-by-$m$). In the general case, the central principle is a restricted version of Generalized Finite Cancellation within Archimedean classes which we call Covered Generalized Finite Cancellation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_03976 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The Logic of Cardinality Comparison Without the Axiom of Choice Harrison-Trainor, Matthew Kulshreshtha, Dhruv Logic 03E10 We work in the setting of Zermelo-Fraenkel set theory without assuming the Axiom of Choice. We consider sets with the Boolean operations together with the additional structure of comparing cardinality (in the Cantorian sense of injections). What principles does one need to add to the laws of Boolean algebra to reason not only about intersection, union, and complementation of sets, but also about the relative size of sets? We give a complete axiomatization. A particularly interesting case is when one restricts to the Dedekind-finite sets. In this case, one needs exactly the same principles as for reasoning about imprecise probability comparisons, the central principle being Generalized Finite Cancellation (which includes, as a special case, division-by-$m$). In the general case, the central principle is a restricted version of Generalized Finite Cancellation within Archimedean classes which we call Covered Generalized Finite Cancellation. |
| title | The Logic of Cardinality Comparison Without the Axiom of Choice |
| topic | Logic 03E10 |
| url | https://arxiv.org/abs/2211.03976 |