Countable models of weakly quasi-o-minimal theories I
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914340312973312 |
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| author | Moconja, Slavko Tanović, Predrag |
| author_facet | Moconja, Slavko Tanović, Predrag |
| contents | We introduce the notions of triviality and order-triviality for global invariant types in an arbitrary first-order theory and show that they are well behaved in the NIP context. We show that these two notions agree for invariant global extensions of a weakly o-minimal type, in which case we say that the type is trivial. In the o-minimal case, we prove that every definable complete 1-type over a model is trivial. We prove that the triviality has several favorable properties; in particular, it is preserved in nonforking extensions of a weakly o-minimal type and under weak nonorthogonality of weakly o-minimal types. We introduce the notion of a shift in a linearly ordered structure that generalizes the successor function. Then we apply the techniques developed to prove that every weakly quasi-o-minimal theory that admits a definable shift has $2^{\aleph_0}$ countable models. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_20589 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Countable models of weakly quasi-o-minimal theories I Moconja, Slavko Tanović, Predrag Logic 03C15 (primary) 03C64 (Secondary) We introduce the notions of triviality and order-triviality for global invariant types in an arbitrary first-order theory and show that they are well behaved in the NIP context. We show that these two notions agree for invariant global extensions of a weakly o-minimal type, in which case we say that the type is trivial. In the o-minimal case, we prove that every definable complete 1-type over a model is trivial. We prove that the triviality has several favorable properties; in particular, it is preserved in nonforking extensions of a weakly o-minimal type and under weak nonorthogonality of weakly o-minimal types. We introduce the notion of a shift in a linearly ordered structure that generalizes the successor function. Then we apply the techniques developed to prove that every weakly quasi-o-minimal theory that admits a definable shift has $2^{\aleph_0}$ countable models. |
| title | Countable models of weakly quasi-o-minimal theories I |
| topic | Logic 03C15 (primary) 03C64 (Secondary) |
| url | https://arxiv.org/abs/2412.20589 |