author={Uri Andrews},
  journal={The Journal of Symbolic Logic},
  pages={186 - 192}
  • U. Andrews
  • Published 1 March 2014
  • Mathematics
  • The Journal of Symbolic Logic
Abstract We characterize the ω-stable theories all of whose countable models admit decidable presentations. In particular, we show that for a countable ω-stable T, every countable model of T admits a decidable presentation if and only if all n-types in T are recursive and T has only countably many countable models. We further characterize the decidable models of ω-stable theories with countably many countable models as those which realize only recursive types. 
There is no classification of the decidably presentable structures
There is no reasonable classification of the decidably presentable structures and it is shown that the index set of the computable structures with decidable presentations is [Formula: see text]-complete.
Algebraic structures computable without delay


A proof of vaught’s conjecture forω-stable theories
In this paper it is proved that ifT is a countable completeω-stable theory in ordinary logic, thenT has either continuum many, or at most countably many, non-isomorphic countable models.
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It is proved that if a decidable complete first order theory has only countably many complete types, then it has a prime model that is almost decidable, and a variation of the open question via a different notion of effectiveness—almost decidable.
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A positive answer is given for the theory of differentially closed fields, and for any decidable ℵ 1 -categorical theory, that a complete countable theory can have at most one prime model up to isomorphism.
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InA proof of Vaught’s Conjecture for ω-stable theories, S. Shelah, L. Harrington and M. Makkai show thatω-stable theories satisfy Vaught’s Conjecture. By using their results and pushing the analysis
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This work will assume that the reader is familiar with Henkin constructions of models, and of passing from a maximal consistent set of sentences, with "Henkin constants", to a model.
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Resplendency and recursive definability inω-stable theories
AbstractWe prove Theorem A.Every resplendent model of an ω-stable theory is homogeneous. As an application we obtain Theorem B.Suppose T is ω-stable, M ⊨ T is recursively saturated and P ∈ S (M) is