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Completeness properties of heyting's predicate calculus with respect to re models

  • Dov M. Gabbay (a1)

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Validity in recursive structures was investigated by several authors. Kreisel [10] has shown that there exists a consistent sentence of classical predicate calculus (CPC) that does not possess a recursive model. The sentence is a conjunction of the axioms of a variant of Bernays set theory, including the axiom of infinity. The language contains additional constants besides ϵ. Later Kreisel [2] and Mostowski [3] presented a sentence (not possessing recursive models) which was a conjunction of axioms of a variant of Bernays set theory without the axiom of infinity but still with additional constants besides ϵ. Later Mostowski [4] improved the result by giving a sentence which can be demonstrated in Heyting arithmetic to be consistent and to have no recursive models. Rabin [6] obtained a simple proof that some sentence of set theory with the single nonlogical constant ϵ does not have any recursively enumerable models.

More generally, Mostowski [5] has shown that the set of all sentences valid in all RE models is not arithmetical and Vaught [1] improved this result by showing that it holds for a language of one binary relation. In fact, Vaught gives· a way of translating n-place relations to 2-place ones that preserves the RE characteristic of the model. For further results pertaining to recursive models see Vaught [1].

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[1]Vaught, R. L., Sentences true in all constructive models, this Journal, vol. 25 (1960), pp. 3953.
[2]Kreisel, G., Note on arithmetic models for consistent formulae of predicate calculus. II, Proceedings ofXIth International Congress of Philosophy, Amsterdam, 1953, pp. 3949
[3]Mostowski, A., On a system of axioms which has no R-E models, Fundamenta Mathematicae, vol. 40 (1953), pp. 5661.
[4]Mostowski, A., A formulae with no R-E models, Fundamenta Mathematicae, vol. 42 (1955), pp. 125140.
[5]Mostowski, A., On recursive models of formalized arithmetic, Bulletin of the Academy of Science III, vol. 5 (1957), pp. 705710.
[6]Rabin, M., On recursively enumerable and arithmetic models of set theory, this Journal, vol. 23 (1958), pp. 408417.
[7]Kreisel, G. and Dyson, , Analysis of Beth semantic construction of intuitionistic logic, Stanford, 1961, Research Report.
[8]Kripke, S., Semantical analysis for intuitionistic logic. I, Formal systems and recursive functions (Crossley, J. and Dummett, M., Editors), North-Holland, Amsterdam, 1965.
[9]Prawitz, O., Ideas and results in proof theory, Proceedings of the 2nd Scandinavian Symposium 1971, North-Holland, Amsterdam.
[10]Kreisel, G., Fundamenta Mathematicae, vol. 37 (1950), pp. 265285.
[11]Kreisel, G., On weak completeness of intuitionistic predicate logic, this Journal, vol. 27 (1962), pp. 139158.
[12]Gabbay, D. M., Sufficient conditions for the undecidability of an intuitionistic theory, this Journal, vol. 37 (1972), pp. 375384.
[13]Tarski, A., Logic, semantics, metamathematics, Oxford, 1956.
[14]Kreisel, G., unpublished appendix to his paper in Infinitistic methods, Warsaw, 1957.
[15]Gabbay, D. M., On Kreise's notion of validity in Post systems, unpublished draft.

Completeness properties of heyting's predicate calculus with respect to re models

  • Dov M. Gabbay (a1)

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