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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2503.20470 |
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| _version_ | 1866916662911959040 |
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| author | Jeon, Hanul Lutz, Patrick Pakhomov, Fedor Walsh, James |
| author_facet | Jeon, Hanul Lutz, Patrick Pakhomov, Fedor Walsh, James |
| contents | Ranking theories according to their strength is a recurring motif in mathematical logic. We introduce a new ranking of arbitrary (not necessarily recursively axiomatized) theories in terms of the encoding power of their $β$-models: $T\prec_βU$ if every $β$-model of $U$ contains a countable coded $β$-model of $T$. The restriction of $\prec_β$ to theories with $β$-models is well-founded. We establish fundamental properties of the attendant ranking. First, though there are continuum-many theories, every theory has countable $\prec_β$-rank. Second, the $\prec_β$-ranks of $\mathcal{L}_\in$ theories are cofinal in $ω_1$. Third, assuming $V=L$, the $\prec_β$-ranks of $\mathcal{L}_2$ theories are cofinal in $ω_1$. Finally, $δ^1_2$ is the supremum of the $\prec_β$-ranks of finitely axiomatized theories. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_20470 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Ranking theories via encoded $β$-models Jeon, Hanul Lutz, Patrick Pakhomov, Fedor Walsh, James Logic Ranking theories according to their strength is a recurring motif in mathematical logic. We introduce a new ranking of arbitrary (not necessarily recursively axiomatized) theories in terms of the encoding power of their $β$-models: $T\prec_βU$ if every $β$-model of $U$ contains a countable coded $β$-model of $T$. The restriction of $\prec_β$ to theories with $β$-models is well-founded. We establish fundamental properties of the attendant ranking. First, though there are continuum-many theories, every theory has countable $\prec_β$-rank. Second, the $\prec_β$-ranks of $\mathcal{L}_\in$ theories are cofinal in $ω_1$. Third, assuming $V=L$, the $\prec_β$-ranks of $\mathcal{L}_2$ theories are cofinal in $ω_1$. Finally, $δ^1_2$ is the supremum of the $\prec_β$-ranks of finitely axiomatized theories. |
| title | Ranking theories via encoded $β$-models |
| topic | Logic |
| url | https://arxiv.org/abs/2503.20470 |