The number of countable models of first-order theories
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arXiv
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| Format: | Preprint |
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2025
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| author | Pillay, Anand Tanović, Predrag |
| author_facet | Pillay, Anand Tanović, Predrag |
| contents | Throughout, $T$ denotes a complete first-order theory in a countable language $L$ that has infinite models and $I(\aleph_0,T)$ denotes the number of countable models of $T$, up to an isomorphism. To determine $I(\aleph_0,T)$, it suffices to consider only countable models of $T$ with domain $ω$; since there are at most continuum many $L$-structures with domain $ω$, $I(\aleph_0,T)\leqslant 2^{\aleph_0}$ holds. Theories with $I(\aleph_0,T)=1$ are the $\aleph_0$-categorical theories. These include the theory of an infinite set, theories of infinite-dimensional vector spaces over a finite field, and the theory of dense linear orders. Theories with $I(\aleph_0,T)<2^{\aleph_0}$ are said to have few countable models.
In this paper we discuss and survey work done on Vaught's conjecture, Martin's conjecture, and Ehhrenfeuch theories (theories with more than one but only finitely many, countable models). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_06854 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The number of countable models of first-order theories Pillay, Anand Tanović, Predrag Logic 03C15 (Primary) 03C45, 03C64 (Secondary) Throughout, $T$ denotes a complete first-order theory in a countable language $L$ that has infinite models and $I(\aleph_0,T)$ denotes the number of countable models of $T$, up to an isomorphism. To determine $I(\aleph_0,T)$, it suffices to consider only countable models of $T$ with domain $ω$; since there are at most continuum many $L$-structures with domain $ω$, $I(\aleph_0,T)\leqslant 2^{\aleph_0}$ holds. Theories with $I(\aleph_0,T)=1$ are the $\aleph_0$-categorical theories. These include the theory of an infinite set, theories of infinite-dimensional vector spaces over a finite field, and the theory of dense linear orders. Theories with $I(\aleph_0,T)<2^{\aleph_0}$ are said to have few countable models. In this paper we discuss and survey work done on Vaught's conjecture, Martin's conjecture, and Ehhrenfeuch theories (theories with more than one but only finitely many, countable models). |
| title | The number of countable models of first-order theories |
| topic | Logic 03C15 (Primary) 03C45, 03C64 (Secondary) |
| url | https://arxiv.org/abs/2508.06854 |