Modal group theory
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866916011920326656 |
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| author | Wołoszyn, Wojciech Aleksander |
| author_facet | Wołoszyn, Wojciech Aleksander |
| contents | I introduce modal group theory, in which we study the category of all groups, considering embeddability as providing a notion of modal possibility. Using HNN extensions and Britton's lemma, I demonstrate that the modal language of groups is more expressive than the first-order language of groups. I interpret the theory of true arithmetic in modal group theory, and show that, as sets of Goedel numbers, it is computably isomorphic to the modal theory of finitely presented groups. I answer an open question of Berger, Block, and Loewe by showing that the formulaic propositional modal validities of groups under embeddings are precisely S4.2. I also analyze sentential validities and worlds validating S5. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_14197 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Modal group theory Wołoszyn, Wojciech Aleksander Logic Group Theory 03B45, 20E06, 20F05, 20F10, 03C57, 03C20 I introduce modal group theory, in which we study the category of all groups, considering embeddability as providing a notion of modal possibility. Using HNN extensions and Britton's lemma, I demonstrate that the modal language of groups is more expressive than the first-order language of groups. I interpret the theory of true arithmetic in modal group theory, and show that, as sets of Goedel numbers, it is computably isomorphic to the modal theory of finitely presented groups. I answer an open question of Berger, Block, and Loewe by showing that the formulaic propositional modal validities of groups under embeddings are precisely S4.2. I also analyze sentential validities and worlds validating S5. |
| title | Modal group theory |
| topic | Logic Group Theory 03B45, 20E06, 20F05, 20F10, 03C57, 03C20 |
| url | https://arxiv.org/abs/2605.14197 |