A “foundational crisis” occurred already in Greek mathematics, brought about by the Pythagorean discovery of incommensurable quantities. It was Eudoxos who provided new foundations, and since then Greek mathematics has been unshakeable. If one reads modern mathematical textbooks, one is normally told that something very similar occurred in modern mathematics. The calculus invented in the seventeenth century had to go through a crisis caused by the use of divergent series. One is told that by the achievements of the nineteenth century from Cauchy to Cantor this crisis has definitely been overcome. It is well known, but it is nevertheless very often not taken seriously into account, that this is an illusion. The so-called ε-δ-definitions of the limit concepts are an admirable achievement, but they are only one step towards the goal of a final foundation of analysis. The nineteenth century solution of the problem of foundations consists of recognizing, in addition to the concept of natural number as the basis of arithmetic, another basic concept for analysis, namely the concept of set. By the inventors of set theory it was strongly held that these sets are self-evident to our intuition; but very soon the belief in their self-evidence was destroyed by the set-theoretic paradoxes. After that, about 1908, the period of axiomatic set theory began. In analogy to geometry there was put forward an uninterpreted system of axioms, a formal system. This, of course, is quite possible. A formal system contains strings of marks; and a special class of these strings, the class of the so-called “theorems”, is inductively defined.

We offer a simple alternative proof of Theorem II, Kleene [1]:

Theorem. For certain primitive recursive -predicates R and S,

We separate the definitions of < o and O:

Definition 1. (1) 1 <o2; (2) for all y, if 1 <oy, then y <o 2v; (3) for all e, if (n)(Ey)T(e, no, y) and (n)(e(no) <oe((n+1)o)), then (n)(e(no) <o 3·5e); (4) for all x, y, z, if x <oy, y <oz, then x <oz; (5) a <ob only as required by (1)–(4).

Definition 2. a ϵ O if and only if 1 <o 2α.

If we write α(2a·3b) = O for a <ob in Conditions (1) to (4) of Definition 1 and take their conjunction, we get an arithmetic predicate P(α). By the Frege-Dedekind device for handling inductive definitions,

Hence, by Theorem 1 (Part 2) of [2], we have a primitive recursive S such that a <o b ≡ (α)(Ey)S(a, b, α, y). By Definition 2,

The classical use of the function quantifier (α) in the above proof is conspicuous. It is not clear that the constructive or intuitionistic content of Kleene's proof (cf. [1] Footnote 40) can be obtained by this method.

In this note we show how Carnap's theory of degree of confirmation (inductive logic) may be constructed upon a very simple and restricted semantical basis. The semantical meta-languages here may either be based upon a relation of multiple denotation or may even be non-translational. To these meta-languages a suitable metric for sentences may be appended. Axioms for such meta-languages are suggested, so that an especially simple formalization of Carnap's inductive logic is in effect presented in outline.

For Carnap, inductive logic consists, roughly speaking, of a semantics augmented by a metric. More specifically, for a finite object-language the semantics consists of a theory of designation for , and to this are added numerical functions on sentences of taking real numbers as values. (, it will be recalled, contains just N distinct individual constants as primitive, where ‘N’ is a constant standing for some fixed finite number. contains a denumerable infinity of distinct individual constants as primitive, and will be discussed in a moment.) Were we to formalize this theory we should have, then, a translation of (or just itself), a syntax for , a primitive relation of designation, real numbers, and several different kinds of class and relational variables. The character of the underlying syntax is not clear from Carnap's account, but this can easily be supplied by a suitable adaptation of Tarski's method. For this, variables for classes of and relations between the expressions of the object-language are needed. Also in Carnap's theory of designation, variables over relations between expressions and objects are needed fundamentally. Real numbers may be handled in various ways. To simplify, let us suppose these are introduced by having a special kind of variable to range over them. Then suitable axioms for real numbers may be given, e.g., one of the axiom sets of Tarski.

A (binary) relation is a collection of ordered couples. Two relations are isomorphic if there is a 1–1 correspondence between their fields which preserves the ordered couples. Isomorphism between relations is itself an equivalence relation, and a structure is an isomorphism class of binary relations.

Carnap ([1], p. 124) asks certain questions concerning (both the exact and) the asymptotic value of the relative frequency that a relation on p objects satisfies certain properties. Among the most interesting special cases are the asymptotic value of the relative frequency that a binary relation on p objects be (a) symmetric, (b) reflexive, (c) transitive irreflexive anti-symmetric, and (d) symmetric irreflexive. These results already appear either in, or almost in, the literature, in various disguises. The object of this expository note is to bring them to light, especially since some of the references may not be generally known to logicians.

Carnap ([1], p. 124) points out explicitly the correspondence between binary relations and linear graphs. Davis [2] and Harary [4] obtains precise results for the number of structures with certain properties using the language of relations and graphs respectively; but these do not supply the asymptotic values directly. The asymptotic results which are most useful here are either contained in, or are straight-forward extensions of, the formulas presented in Ford and Uhlenbeck [3], which in turn are based on unpublished work of G. Pólya. In addition, we utilize a result of L. Moser on transitive relations, which appears in Wine and Freund [5].

What I am offering here is a reconsideration, and in the end a solution, of the ancient paradox of the Epimenides. What has provoked this new assault on so old a stronghold is L. Jonathan Cohen's [2], in which it is rightly pointed out that the Epimenidean as contrasted with the Eubulidean version of the Liar paradox is the one that threatens logicians who attempt to formalise the use of indirect rather than direct discourse. Among these he includes myself, referring to the system sketched on pp. 130–131 of my [4], and suggesting that I ought to have shown how a person interpreting this system in the obvious way can avoid semantic antinomies. Here too he is right, and my main purpose now is not to criticise his paper, but to fill in this lacuna in my own work to which he has drawn attention. I do want, however, to make one small criticism at the start, namely that Cohen fails to notice what it is that is really paradoxical about the Epimenides, and in consequence fails to perceive the same paradoxical character in a proposition which he is himself quite happy to accept as a ‘logical truth.’

The point about the Epimenides, as was noted by Church in [1], is not that when we examine the truth value of a Cretan's assertion that nothing true is ever asserted by a Cretan we are led to contradictory conclusions, for we are not; the paradox is rather that such examination makes it seem possible to settle an empirical question on logical grounds. If we treat the Cretan's assertion as true, and so assume that nothing true is ever asserted by a Cretan, it follows immediately that the Cretan's assertion is false. If, however, we treat it as false, there is no way of deducing from this assumption that it is true. We can, therefore, consistently suppose it to be false, and this is all that we can consistently suppose. But to suppose it false (considering what the assertion actually is) is to suppose that something asserted by a Cretan is true; and this of course can only be some other assertion than the one mentioned. We thus reach the peculiar conclusion that if any Cretan does assert that nothing asserted by a Cretan is true, then this cannot possibly be the only assertion made by a Cretan — there must also be, beside this false Cretan assertion, some true one.

In [2] E. L. Post defined a many-valued propositional logic to be functionally complete if, for every function on the set of truth-values, there exists a formula of the logic having that function as its associated truth-value function. He proved that the logic with truth-values 1, 2, …, m and (i) a unary connective ∼ such that ∼p has truth-value i+1 (mod m) when p has truth-value i, (ii) a binary connective ∨ such that p ∨ q has truth-value min(i, j) when p, q have truth-values i, j respectively, is functionally complete.

The many-valued logics described by Łukasiewicz and Tarski [1] are not functionally complete. These logics have truth-values 1, 2, …, m and (i) a unary connective ~ such that ~p has truth-value m−i+1 when p has truth-value i, (ii) a binary connective → such that if p, q have truth-values i, j respectively, then p → q has truth-value 1 for i ≧ j, and truth-value 1 for i ≧ j. The functional incompleteness of these logics is immediate, since there exists no formula in p having truth-value i (≠ 1 or m) when p has truth-value 1.

In [4] Słupecki showed that if a new unary connective T, such that T(p) has truth-value 2 for all truth-values assigned to p, is added to the 3-valued Łukasiewicz-Tarski logic, then the resulting logic is functionally complete. In [3] Rosser and Turquette proved this result for the m-valued (m ≧ 3) logic.

In this paper an affirmative solution is given for the following problem in many-valued logic posed by Rosser and Turquette in [6] p. 110:

For every triple ‹s, t, m› with 1 ≤ s < t < m, is it possible to define a system of m-valued logic which satisfies the following conditions? I. Every statement which always takes values ≦ s is provable. II. No statement which ever takes a value > t is provable. III. Of those statements which always take values ≤ t and sometimes take a value > s, some are provable and some are not.

We state now, temporarily deferring comment, a more demanding alternative to Condition III: III′. For every truth value k such that s < k ≦ t, there are statements P and Q such that each takes the value k at least once, but never takes a value > k, and P is provable and Q is not.

In § 2 a solution is given on the level of the statement calculus. The remainder of the present section is devoted to some negative results which serve to clarify the problem, and one positive result which shall cause us to replace Condition III by III′. We confine our attention to the statement calculus.

Kleene-Post [10], Spector [14] and Friedberg [3, 4], have given a number of examples of functions which have incomparable degrees of recursive unsolvability satisfying various additional conditions. Our present purpose is to determine to what extent some of these incomparability theorems can be generalized by substituting for “α. is recursive in β” other relations Q(α, β) such as “α is hyperarithmetical in β” (which we shall think of as “α ≦ β”), and to determine what restrictions, if any, need be imposed on Q. As a consequence of our investigation we shall show that there are incomparable hyperdegrees as defined in [9] and [13].

By a theory we shall always mean one of first order, having finitely many non-logical constants. Then for theories with identity (as a logical constant, the theory being closed under deduction in first-order logic with identity), and also likewise for theories without identity, one may distinguish the following three notions of axiomatizability. First, a theory may be recursively axiomatizable, or, as we shall say, simply, axiomatizable. Second, a theory may be finitely axiomatizable using additional predicates (f. a.+), in the syntactical sense introduced by Kleene [9]. Finally, the italicized phrase may also be interpreted semantically. The resulting notion will be called s. f. a.+. It is closely related to the modeltheoretic notion PC introduced by Tarski [16], or rather, more strictly speaking, to PC∩ACδ.

For arbitrary theories with or without identity, it is easily seen that s. f. a.+ implies f. a.+ and it is known that f. a.+ implies axiomatizability. Thus it is natural to ask under what conditions the converse implications hold, since then the notions concerned coincide and one can pass from one to the other.

Kleene [9] has shown: (1) For arbitrary theories without identity, axiomatizability implies f. a.+. It also follows from his work that : (2) For theories with identity which have only infinite models, axiomatizability implies f. a.+.

In this paper we shall prove three theorems about recursively enumerable sets. The first two answer questions posed by Myhill [1].

The three proofs are independent and can be presented in any order. Certain notations will be common to all three. We shall denote by “Re” the set enumerated by the procedure of which e is the Gödel number. In describing the construction for each proof, we shall suppose that a clerk is carrying out the simultaneous enumeration of R0, R1, R2, …, in such a way that at each step only a finite number of sets have begun to be enumerated and only a finite number of the members of any set have been listed. (One plan the clerk can follow is to turn his attention at Step a to the enumeration of Re where e+1 is the number of prime factors of a. Then each Re receives his attention infinitely often.) We shall denote by “Rea” the set of numbers which, at or before Step a, the clerk has listed as members of Re. Obviously, all the Rea are finite sets, recursive uniformly in e and a. For any a we can determine effectively the highest e for which Rea is not empty, and for any a, e we can effectively find the highest member of Rea, just by scanning what the clerk has done by Step a. Additional notations will be introduced in the proofs to which they pertain.

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.