Sufficient conditions for the undecidability of intuitionistic theories with applications

Abstract

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′.

The undecidability of intuitionistic theories of algebraically closed fields and real closed fields

“…Is T H decidable (or, more generally, if X is any intermediate logic, is T x decidable)? In [1] we gave sufficient conditions for the undecidability of T H . These conditions depend on the formulas of T (different axiomatization of the same T c may give rise to different T H ) and on the classical model theoretic properties of T° (the method did not work for model complete theories, e.g.…”

“…those of the title of the paper). For details see [1]. In [2] we gave some decidability results for some theories: The problem of the decidability of theories T H for a classically model complete T c remained open.…”

“…The cases of algebraically closed fields and real closed fields and divisible abelian groups are treated in this paper. Other various decidability results of the intuitionistic theories were obtained by several authors, see [1], [2], [4] for details.…”

“…The formulations that we consider here are of interest as these systems occur in intuitionistic mathematics. 1 We hope that the present methods could be extended to the (more interesting) case of Heyting's systems [5]. §1.…”

“…(4) Unary function (one), etc. In [1], the HPC theory of abelian groups and unary function with decidable equality were shown undecidable. This result implies (2) and 4 Again, the following holds: T H (or 7*0 h 5* iffRSh-S.…”