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# Decidability of some intuitionistic predicate theories

## Extract

Suppose T is a first order intuitionistic theory (more precisely, a theory of Heyting's predicate calculus, e.g., abelian groups, one unary function, dense linear order, etc.) presented to us by a set of axioms (denoted also by) T.

Question. Is T decidable?

One knows that if the classical counterpart of T (i.e., take the same axioms but with the classical predicate calculus as the underlying logic) is not decidable, then T cannot be decidable. The problem remains for theories whose classical counterpart is decidable. In [8], sufficient conditions for undecidability were given, and several intuitionistic theories such as abelian groups and unary functions (both with decidable equality) were shown to be undecidable. In this note we show decidability results (see Theorems 1 and 2 below), and compare these results with the undecidability results previously obtained. The method we use is the reduction-method, described fully in [12] and widely applied in [3], which is applied here roughly as follows:

Let T be a given theory of Heyting's predicate calculus. We know that Heyting's predicate calculus is complete for the Kripke-model type of semantics. We choose a class M of Kripke models for which T is complete, i.e., all axioms of T are valid in any model of the class and whenever φ is not a theorem of T, φ is false in some model of M.

## References

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[1]Rabin, M. O., Decidability of second order theories and automata on trees, Transactions of the American Mathematical Society, vol. 141 (1969), pp. 135.
[2]Lifshits, V. A., Problems of decidability for some constructive theories of equality, Studies in constructive mathematics and mathematical logic (Slisenko, O., Editor), Consultants Bureau, 1969.
[3]Gabbay, D. M., Decidability results in non-classical logics, Technical Report, Office of Naval Research, Jerusalem, 1969.
[4]Smorynski, C., The undecidability of some intuitionistic theories of equality and order, Stanford University (unpublished).
[5]Maslov, S. Y., Minc, G. B. and Orevkov, V. P., Insolvability in the constructive predicate calculus of certain classes of formulas containing only monadic predicate variables, Doklady Akademii Nauk SSSR, vol. 163 (1965), pp. 295297; Soviet Mathematics Doklady, vol. 6 (1965), pp. 918–920.
[6]Gabbay, D. M., The undecidability of intuitionistic theories of algebraic closed fields and real closed fields, this Journal (to appear).
[7]Kripke, S. R., Semantic analysis for intuitionistic logic. II (unpublished).
[8]Gabbay, D. M., Sufficient conditions for the undecidability of intuitionistic theories, this Journal, vol. 37 (1972), pp. 375384.
[9]Rogers, H., Certain logical reduction and decision problems, Annals of Mathematics, vol. 64 (1956), pp. 264284.
[10]Horn, A., Logic with truth values in a linearly ordered Heyting algebra, this Journal, vol. 34 (1969), pp. 395408.
[11]Kripke, S. R., Semantic analysis for intuitionistic logic. I, Formal systems and recursive functions (Crossley-Dummet, , Editors), North-Holland, Amsterdam.
[12]Gabbay, D. M., Survey of decidability results in non-classical logics, Proceedings of the fourth international congress in logic, methodology and the philosophy of science, North-Holland, Amsterdam (to appear).

# Decidability of some intuitionistic predicate theories

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