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Sufficient conditions for the undecidability of intuitionistic theories with applications1

  • Dov M. Gabbay (a1)

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Let Δ be a set of axioms of a theory Tc(Δ) of classical predicate calculus (CPC); Δ may also be considered as a set of axioms of a theory TH(Δ) of Heyting's predicate calculus (HPC). Our aim is to investigate the decision problem of TH(Δ) in HPC for various known theories Δ of CPC.

Theorem I(a) of §1 states that if Δ is a finitely axiomatizable and undecidable theory of CPC then TH(Δ) is undecidable in HPC. Furthermore, the relations between theorems of HPC are more complicated and so two CPC-equivalent axiomatizations of the same theory may give rise to two different HPC theories, in fact, one decidable and the other not.

Semantically, the Kripke models (for which HPC is complete) are partially ordered families of classical models. Thus a formula expresses a property of a family of classical models (i.e. of a Kripke model). A theory Θ expresses a set of such properties. It may happen that a class of Kripke models defined by a set of formulas Θ is also definable in CPC (in a possibly richer language) by a CPC-theory Θ′! This establishes a connection between the decision problem of Θ in HPC and that of Θ′ in CPC. In particular if Θ′ is undecidable, so is Θ. Theorems II and III of §1 give sufficient conditions on Θ to be such that the corresponding Θ′ is undecidable. Θ′ is shown undecidable by interpreting the CPC theory of a reflexive and symmetric relation in Θ′.

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1

This research has been supported in part by National Science Foundation grant GJ-443X. I am indebted to Professor Kreisel for very helpful criticism. All possible shortcomings are entirely my responsibility.

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[1]Maslov, S. Y., Minc, G. B. and Orevkov, V. P., Unsolvability of the constructive predicate calculus of certain classes of formulas containing only monadic predicate variables, Soviet Mathematics Doklady, vol. 163 (1965), pp. 918920.
[2]Slomson, A. B., An undecidable two sorted predicate calculus, this Journal, vol. 34 (1969), pp. 2123.
[3]Kripke, S. A., Semantic analysis for intuitionistic logic. II (mimeographed).
[4]Kripke, S. A., The undecidability of monadic modal quantification theory, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 8 (1962), pp. 113116.
[5]Orevkov, V. P., The undecidability of a class of formulas containing just one single place predicate variable in modal calculus, Studies in constructive mathematics and mathematical logic, New York, 1969, pp. 6770.
[6]Smorynski, C., Some recent results on elementary intuitionistic theories, Stanford University, California (mimeographed).
[7]Gabbay, D. M., Decidability of some intuitionistic predicate theories this Journal (to appear).
[8]Kripke, S. A., Semantic analysis for intuitionistic logic. I, Formal systems and recursive functions, North-Holland, Amsterdam 1965.
[9]Rogers, H., Certain logical reduction and decision problems, Annals of Mathematics, vol. 64 (1956), pp. 264284.
[10]Gabbay, D. M., Applications of trees to intermediate logics, this Journal, vol. 37 (1972) pp. 135138.
[11]Lifshits, V. A., Problem of decidability for some constructive theories of equality, Studies of constructive mathematics and mathematical logic, New York, 1969.
[12]Gabbay, D. M., The undecidability of intuitionistic theories of algebraic and real closed fields, Stanford University, California (mimeographed).

Sufficient conditions for the undecidability of intuitionistic theories with applications1

  • Dov M. Gabbay (a1)

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