Abstract
In this paper we obtain two results using amalgamation classes and Fraïssé limits. First, we construct a decidable theory T whose types are all decidable yet whose prime model is not decidable. Millar [15] constructed such example but his example uses an infinite language in an essential way. Our example uses one binary predicate symbol, that is, the models we construct are graphs. Second, for any finite lattice \(\cal F\) we construct a theory T with countably many models such that the fundamental order determined by T is isomorphic to \(\cal F\). As a by-product of this example, we propose the investigation of computable and decidable models of T by connecting them to the fundamental order of T.
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Gavruskin, A., Khoussainov, B. (2013). On Decidable and Computable Models of Theories. In: Bonizzoni, P., Brattka, V., Löwe, B. (eds) The Nature of Computation. Logic, Algorithms, Applications. CiE 2013. Lecture Notes in Computer Science, vol 7921. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39053-1_23
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DOI: https://doi.org/10.1007/978-3-642-39053-1_23
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