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Homogeneous models and almost decidability

Part of: Model theory

Published online by Cambridge University Press:  09 April 2009

Terry Millar
Terry Millar
Affiliation:
Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706, U.S.A.
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Abstract

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Countable homogeneous models are ‘simple’ objects from a model theoretic point of view. From a recursion theoretic point of view they can be complex. For instance the elementary theory of such a model might be undecidable, or the set of complete types might be recursively complex. Unfortunately even if neither of these conditions holds, such a model still can be undecidable. This paper investigates countable homogeneous models with respect to a weaker notion of decidability called almost decidable. It is shown that for theories that have only countably many type spectra, any countable homogeneous model of such a theory that has a Σ2 type spectrum is almost decidable.

MSC classification

Secondary: 03C57: Effective and recursion-theoretic model theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 46 , Issue 3 , June 1989 , pp. 343 - 355
Copyright
Copyright © Australian Mathematical Society 1989

References

[1]Millar, T., ‘Prime models and almost decidability’, J. Symbolic Logic 51 (1986), 412–420.CrossRefGoogle Scholar
[2]Millar, T., ‘Type structure complexity and decidability’, Trans. Amer. Math. Soc. 271 (1981), 73–81.CrossRefGoogle Scholar
[3]Millar, T., ‘Homogeneous models and decidability’, Pacific J. Math. 91 (1980), 407–418.CrossRefGoogle Scholar
[4]Millar, T., ‘Decidability and the number of countable models’, Ann. Pure Appl. Logic 27 (1984), 137–153.CrossRefGoogle Scholar