Categoricity without Power
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914524506882048 |
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| author | Goh, Jun Le Tran, Chieu-Minh |
| author_facet | Goh, Jun Le Tran, Chieu-Minh |
| contents | We prove an analogue of Morley's categoricity theorem where cardinality is replaced by the recursion-theoretic notion of arithmetic degree. We say that a complete arithmetically definable theory $T$ is $D$-categorical if any two arithmetically extendible models of $T$ of arithmetic degree $D$, considered over a common elementary submodel with arithmetical elementary diagram, are isomorphic over that submodel by an isomorphism which preserves the complexity of sets of degree $D$. Here an arithmetically extendible model means an elementary substructure of a model whose elementary diagram is arithmetical. Our main result is: If $T$ is $D_1$-categorical for some nonzero arithmetic degree $D_1$, then $T$ is $D_2$-categorical for every nonzero arithmetic degree $D_2$. We also show that, assuming ZFC, $D$-categoricity for some nonzero arithmetic degree is equivalent to uncountable categoricity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_00697 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Categoricity without Power Goh, Jun Le Tran, Chieu-Minh Logic We prove an analogue of Morley's categoricity theorem where cardinality is replaced by the recursion-theoretic notion of arithmetic degree. We say that a complete arithmetically definable theory $T$ is $D$-categorical if any two arithmetically extendible models of $T$ of arithmetic degree $D$, considered over a common elementary submodel with arithmetical elementary diagram, are isomorphic over that submodel by an isomorphism which preserves the complexity of sets of degree $D$. Here an arithmetically extendible model means an elementary substructure of a model whose elementary diagram is arithmetical. Our main result is: If $T$ is $D_1$-categorical for some nonzero arithmetic degree $D_1$, then $T$ is $D_2$-categorical for every nonzero arithmetic degree $D_2$. We also show that, assuming ZFC, $D$-categoricity for some nonzero arithmetic degree is equivalent to uncountable categoricity. |
| title | Categoricity without Power |
| topic | Logic |
| url | https://arxiv.org/abs/2605.00697 |