The purpose of the present article is to formulate in an intuitionistically meaningful manner the completeness problems for the intuitionistic predicate calculus, and to establish the completeness of certain fragments of it. In these proofs certain translations of classical logic into intuitionistic logic are used, in particular those discovered by Kolmogorov [11] and Gödel [3], and a new one in which negations of prenex formulae are central. Since the last one is of interest also independently of the completeness problems, the details are presented separately at the end of this paper.

All arguments are intended to be intuitionistically valid unless the opposite is stated; in particular, ‘proof’ means intuitionistic proof, and ‘formal proof’ means proof in the relevant system of Heyting [4], [5].

Open problems are mentioned in Remarks 3.1 and 8.1.

German capitals denote formulae of predicate logic, Latin (italic) capitals denote predicate symbols or propositional letters.

In § 1 we present conceptual material concerning the notion of a Gödel numbering of the partial recursive functions. § 2 presents a theorem about these concepts. § 3 gives several applications. The material in § 1 and § 2 grew out of attempts by the author to find routine solutions to some of the problems discussed in § 3. The author wishes to acknowledge his debt in § 2 to the fruitful methods of Myhill in [M] and to thank the referee for an abbreviated and improved version of the proof for Lemma 3 in § 2.

In the literature of mathematical logic, “Gödel numbering” usually means an effective correspondence between integers and the well-formed formulas of some logical calculus. In recursive function theory, certain such associations between the non-negative integers and instructions for computing partial recursive functions have been fundamental. In the present paper we shall be concerned only with numberings of the latter, more special, sort. By numbers and integers we shall mean non-negative integers. Our notation is, in general, that of [K]. If ϕ and ψ are two partial functions, ϕ = ψ shall mean that (∀x)[ϕ(x)≃(ψx)], i.e., that ϕ and ψ are defined for the same arguments and are equal on those arguments. We consider partial recursive functions of one variable; applications of the paper to the case of several variables, or to the case of all partial recursive functions in any number of variables, can be made in the usual way using the coordinate functions (a)i of [K, p. 230]. It will furthermore be observed that we consider only concepts that are invariant with respect to general recursive functions; more limited notions of Gödel numbering, taking into account, say, primitive recursive structure, are beyond the scope of the present paper.