The present work was inspired by Mostowski's paper [3] in which he considers classes of sequences of real numbers, and associated sets of primitive recursive real numbers. The present treatment will differ from that of [3] in that we consider the real numbers in (0, ∞), instead of just those in (0, 1), but this will make little difference. In §1 we develop a general theory, and in §2 we show the relation of this theory to the results contained in [3], and to those of other authors.

This note will characterize those partial functions which are partial recursive under an arbitrary “renaming” of the integers.

The formal result. Let I be the non-negative integers. For any partial function f on I into I, and for any permutation σ of I, let fσ be the partial function which f becomes when the integers are renamed according to σ. That is, let fσ = σfσ−1 in the sense that fσ(x) is defined whenever σ−1(х) ∊ Dom f.

As is well known, some paradoxes arise through inadequate analysis of the meanings of terms in a language, an adequate analysis showing that the paradoxes arise through a lack of separation of an object theory and a metatheory. Under such an adequate analysis in which parts of the metatheory are modelled in the object theory, the paradoxes give way to remarkable theorems establishing limitations of the object theory.

Such a modelling is often accomplished by a Gödel numbering. Here we shall use a somewhat different technique in axiomatic set theory, from which we shall reap a few results having the effect of comparing the strength of various axiom schema of comprehension for sets and classes (cf. the numbered results of §§5–7). Similar results were obtained by A. Mostowski [7] using Gödel numbering (cf. 5.3 and 7.3 below).

In this note we partly answer a question left open in [1, p. 52] by proving the following theorem.

Theorem. Supposep0≧2, ωn+1·p0≦σ<ωn+1(p0+1). Then W(σ) is uniformly many-one reducible to W(τ) for τ≧(ωn+1·p0)+1.

To prove this theorem we need only show that there is a recursive function f(e) such that if ∣e∣<σ, then ∣f(e)<(ωn+1·p0)+1, while if ∣e∣ ≮ σ, f(e)∉W. (For the notations see [2].)

This paper contains some results concerning the completeness of a first-order system of infinite valued logic

There are under consideration two distinct notions of completeness corresponding to the two notions of validity (see Definition 3) and strong validity (see Definition 4). Both notions of validity, whether based on the unit interval [0, 1] or based on linearly ordered MV-algebras, use the element 1 as the designated truth value. Originally, it was thought by many investigators in the field that one should be able to prove that the set of valid sentences is recursively enumerable. It was first proved by Rutledge in [9] that the set of valid sentences in the monadic first-order infinite valued logic is recursively enumerable.

C. I. Lewis intended his systems S1–S5 as contributions to the study of “strict implication”, but in his formulation, strict implication is so thoroughly intertwined with other notions, such as possibility and negation, that it remains a problem, to separate out the properties of strict implication itself. I shall solve this problem for S2–5 and von Wright's M. The results for S3–5 are given below, while the implicative parts of S2 and M, which are rather more complicated, are given in §5.

In this presentation, ‘⊃’ stands for strict implication, and missing brackets for ‘⊃’ are restored by association to the right. A strict formula is one of the form A ⊃ B. Axiom schemes are used throughout.

A system formalized within the first order predicate calculus either with or without the identity but without predicate or function variables we call simply a theory. We say that a theory T has a recursively enumerable complement (r.e. Comp.) if the set of all sentences of T not valid in T is recursively enumerable. This is equivalent to saying that there exists an effective, purely mechanical procedure for establishing the non-validity in T of precisely those sentences of T which are in fact not theorems of T. If in addition T is recursively enumerable, that is, if there is an effective procedure for establishing the validity in T of its theorems, then T is decidable. It will be shown here that several well-known results concerning decidable theories can be extended to cover theories with r.e. comps. In this respect it should be observed that there exist theories having r.e. comps. which are not decidable. An example is the theory which has as valid sentences just those which contain a given two-place predicate and which are valid in all non-void finite universes [3], [4].

In [1], Bemays gives an axiomatization of set-theory which represents the furthest development to date of the Zermelo-Frankel-Gödel tradition. The purpose of this note is to show that one of his axioms is redundant.

Primitives are ∈, set-abstraction (denoted [x/A(x)]) and the Hilbert selector (denoted σ). Different styles are used for free variables (a, b,c, …) and bound variables (x, y, z, …); both kinds of variables range over both sets and classes. Quantifiers and connectives are classical.

The infinite-valued statement calculus to which this paper refers is that of Łukasiewicz [10], whose axiomatization was proved complete in [5]. In [9], Rutledge extended this system to include predicates and quantifiers2 and presented a deductively complete set of axioms for the monadic predicate calculus. This paper represents an attempt to axiomatize the full predicate calculus; for the proposed axiomatization, a property akin to but weaker than completeness is proved. An attempt to prove full completeness along similar lines failed; it has since been shown [11] that the set of valid formulas of the infinite-valued predicate calculus is not recursively enumerable. The method of this paper was suggested by Professor J. Barkley Rosser.

The system of combinatory logic to be presented in this paper will be called CΔ. It may be compared to various systems of combinatory logic constructed by Schonfinkel [11], Curry [1], Rosser [10], and Curry and Feys [2], and to the author's systems K′ [3] and S [5], and to extensional modifications of K′ [6], [7], [8], [9]. The present system falls within this latter category of extensional or semi-extensional systems, but it is more perspicuous than the others, and the proof of its consistency is more direct. It contains the theory of combinators in full strength. It also contains operators for disjunction, conjunction, negation, existence (that is, non-emptiness), and universality. A considerable part of classical mathematical analysis can be shown to be derivable in CΔ, just as in K′ [4], but with the added advantage of the availability of a limited extensionality principle.