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Countable models of 1-based theories

Anand Pillay

Archive for Mathematical Logic 31 (3):163-169 (1992)

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Fields with few types.Cédric Milliet - 2013 - Journal of Symbolic Logic 78 (1):72-84.details According to Belegradek, a first order structure is weakly small if there are countably many $1$-types over any of its finite subset. We show the following results. A field extension of finite degree of an infinite weakly small field has no Artin-Schreier extension. A weakly small field of characteristic $2$ is finite or algebraically closed. A weakly small division ring of positive characteristic is locally finite dimensional over its centre. A weakly small division ring of characteristic $2$ is a field. Download Export citation Bookmark
An omitting types theorem for saturated structures.A. D. Greif & M. C. Laskowski - 1993 - Annals of Pure and Applied Logic 62 (2):113-118.details We define a new topology on the space of strong types of a given theory and use it to state an omitting types theorem for countably saturated models of the theory. As an application we show that if T is a small, stable theory of finite weight such that every elementary extension of the countably saturated model is ω-saturated then every weakly saturated model is ω-saturated. Download Export citation Bookmark
(1 other version)Countable models of trivial theories which admit finite coding.James Loveys & Predrag Tanović - 1996 - Journal of Symbolic Logic 61 (4):1279-1286.details We prove: Theorem. A complete first order theory in a countable language which is strictly stable, trivial and which admits finite coding has 2 ℵ 0 nonisomorphic countable models. Combined with the corresponding result or superstable theories from [4] our result confirms the Vaught conjecture for trivial theories which admit finite coding. Download Export citation Bookmark 2 citations
Triviality, NDOP and stable varieties.B. Hart, A. Pillay & S. Starchenko - 1993 - Annals of Pure and Applied Logic 62 (2):119-146.details We study perfectly trivial theories, 1-based theories, stable varieties, and their mutual interaction. We give a structure theorem for the models of a complete perfectly trivial stable theory without DOP: any model is the algebraic closure of a nonforking regular tree of elements. We also give a structure theorem for stable varieties, all of whose completions have NDOP. Such a variety is a varietal product of an affine variety and a combinatorial variety of an especially simple form. Download Export citation Bookmark
A note on countable models of 1-based theories.Predrag Tanovic - 2002 - Archive for Mathematical Logic 41 (7):669-671.details We prove that the existence of a nonisolated type having a finite domain and which is orthogonal to øin a 1-based theory implies that it has a continuum nonisomorphic countable models. Download Export citation Bookmark

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