In investigations of the foundations of mathematics we can distinguish two separate tendencies. On the one hand, one may seek to define his subject with greatest possible explicitness: to obtain a formulation which satisfies the most exacting demands for precision, and which is at the same time free from paradoxes and adequate for the purpose. On the other hand, besides the problem of formulation, there is that of simplification; one can seek to find systems based upon processes of greater and greater primitiveness. The reduction of a piece of mathematics to a formal system, and still further to a completely formalized system (as explained, for example, in my New York address a year ago), is a step toward the first of these objectives. But it is evident that one can formalize in various ways, and that some of these ways constitute a more profound analysis than others. Although from some points of view one way of formalization is as good as any other, yet a certain interest attaches to the problem of simplification, as is shown by the attention which some of the greatest mathematicians have devoted to it.

The researches about which I am reporting today are directed toward the second of these objectives. In fact we are concerned with constructing systems of an extremely rudimentary character, which analyze processes ordinarily taken for granted. This is properly part of the business of mathematical logic. Of course there are those, even among logicians, who doubt the utility of this sort of thing—who profess to have no interest in improvements which do not lead to increases in deductive power or what not. However that may be, this second objective certainly has some interest in its own right; and it is this interest which has formed the primary motivation for these researches.

This paper is concerned with finding a fairly simple system of logic which is “basic” in the sense that every system of logic is definable in it. If a “system”is regarded as being a class of propositions, rather than a class of sentences, then every class of propositions which is a system should be definable in the basic logic. The system of logic proposed in this paper will not be proved to be a basic logic, but strong evidence that it is basic will be given at the end of the paper. Evidence will also be given that the class of propositions which are not provable in the system is not definable in any system of logic. It will be established that the decision problem is unsolvable for the system. Notable characteristics of the system are its lack of negation and universal quantification, and its similarity to systems proposed by Church, Curry, and Rosser.

By a “system” will be meant a class of propositions the membership of which can be specified by ordinary recursive methods, so that if the membership of a given proposition in the class can be established at all, such membership can be established in a finite number of steps. (This is not the same as demanding that a criterion must exist for determining, in a finite number of steps, whether or not some given proposition is a member of the class. Such a demand would require a solution of the decision problem for every system.)

Our task in the treatment of general set theory will be to give a survey for the purpose of characterizing the different stages and the principal theorems with respect to their axiomatic requirements from the point of view of our system of axioms. The delimitation of “general set theory” which we have in view differs from that of Fraenkel's general set theory, and also from that of “standard logic” as understood by most logicians. It is adapted rather to the tendency of von Neumann's system of set theory—the von Neumann system having been the first in which the possibility appeared of separating the assumptions which are required for the conceptual formations from those which lead to the Cantor hierarchy of powers. Thus our intention is to obtain general set theory without use of the axioms V d, V c, VI.

It will also be desirable to separate those proofs which can be made without the axiom of choice, and in doing this we shall have to use the axiom V*—i.e., the theorem of replacement taken as an axiom. From V*, as we saw in §4, we can immediately derive V a and V b as theorems, and also the theorem that a function whose domain is represented by a set is itself represented by a functional set; and on the other hand V* was found to be derivable from V a and V b in combination with the axiom of choice. (These statements on deducibility are of course all on the basis of the axioms I–III.)

In the system presented by Quine in his book Mathematical logic, one can derive the Burali-Forti Paradox. It is the purpose of this paper to present the details of this derivation. For the derivation, only familiarity with Quine's book is assumed.

The present derivation is based on the derivations given by Hobson and by Whitehead and Russell.

In both these sources, the primary interest is the theory of ordinals, and the Burali-Forti Paradox is of interest only as something to be avoided. In the present paper the primary interest is the Burali-Forti Paradox, so that much of the theory of ordinals is absent from this paper and only those details remain which are relevant to the derivation of the paradox.

Any logical system, if its use is to be carried beyond a rather elementary stage, needs powerful conventions about abbreviations: in particular one usually wants to modify the bracketing so as to make the formulae more readable, and also possibly shorter. The present note has been written in the belief that Church's formulation of the simple theory of types is particularly suitable as a basis for work on that theory, and that it is therefore worth while introducing special conventions which take into account the needs of this particular system. The conventions which I shall describe are ones which I have used a good deal myself, and have always found adequate. I intend to make use of them in forthcoming papers. They may be regarded as an extension of Curry's conventions.

I shall begin with a general discussion of punctuation by means of groups of dots. This general theory is applicable, with some modifications, to Russell's, Quine's, and Curry's bracketing systems as well as to the present one.

Sets of postulates for at least three different combinatory logics have been given ([2], [3], [4], [5], [6], [7]). The logics and their sets of postulates are quite similar. In all cases it is the proof of the completeness of the set of postulates which causes the difficulty and makes the main papers so long ([3], [7]). It is the purpose of this paper to present a method of writing down sets of postulates for combinatory logics for which the proof of completeness will be fairly simple. We assume acquaintance with certain descriptive portions of [1], namely sections 1–6, 12, 13, 15. Otherwise the present paper is self-contained.

We first solve our problem for the particular logic studied by Rosser ([7]) and then indicate how the method would apply to other logics.

We start with a set, s, of primitive terms. We define s-combination as in [1], p. 43. The precise contents of s need not concern us. It suffices that there be s-combinations I, J, and Q, with no variables in them, and having properties as described below. I and J are to have the same meaning as in [1], p. 43, namely IA is the same as A and JABCD is the same as AB(ADC). QAB is to have the meaning that A and B, considered as functions, are the same.

A proof that certain systems of formal logic are inconsistent, in the sense that every formula expressible in them is provable, was published by Kleene and Rosser under the above title in 1935. For the case where the underlying system satisfies an additional condition—viz. the possession of an operator analogous to Schönfinkel's K—I gave a simpler derivation of this condition in a previous paper. That argument, like the original one of Kleene and Rosser, was a refinement of the Richard paradox. The object of the present paper is to show that, if we use other paradoxes, an inconsistency will result from a very much simpler argument and on much less restrictive hypotheses. The contradiction can no longer be called “the paradox of Kleene and Rosser,” because it is based on an entirely different principle; but, in deference to the work of the original discoverers of the inconsistency, the paper is given the same title as that which their paper bears. The central idea of the new derivation was suggested by some work of R. Carnap.

This paper is based on the one above cited, which will be referred to as PKR. However, acquaintance with that paper will be presupposed only through 3.4, i.e., through the statement of the basic hypotheses and conventions for a combinatorially complete system—except that for the second (alternative) method of construction given below the substance of PKR through 9.6 is needed.

The foundation of analysis does not require the full generality of set theory but can be accomplished within a more restricted frame. Just as for number theory we need not introduce a set of all finite ordinals but only a class of all finite ordinals, all sets which occur being finite, so likewise for analysis we need not have a set of all real numbers but only a class of them, and the sets with which we have to deal are either finite or enumerable.

We begin with the definitions of infinity and enumerability and with some consideration of these concepts on the basis of the axioms I—III, IV, V a, V b, which, as we shall see later, are sufficient for general set theory. Let us recall that the axioms I—III and V a suffice for establishing number theory, in particular for the iteration theorem, and for the theorems on finiteness.

This note is concerned with the logical formalism with types recently introduced by Church [1] (and called (C) in this note) It was shewn in his paper (Theorem 26α) that if Yα stands for

(a form of the “axiom of infinity” for the type α), Yα can be proved formally, from Yι and the axioms 1 to 7, for all types α of the forms ι′, ι″, …. For other types the question was left open, but for the purposes of an intrinsic characterisation of the Church type-stratification given by one of us, it is desirable to have the remaining cases cleared up. A formal proof of Yα is now given for all types α containing ι, but the proof uses, in addition to Axioms 1 to 7 and Yι, also Axiom 9 (in connection with Def. 4), and Axiom 10 (in Theorem 9).

In the middle of my Mathematical logic I defined a certain class of formulae as “stratified,” and conjectured that exclusion from this class is a feature “shared, presumably, by all the untenable statements (p. 157). This ushered in a set of axioms of class-membership which Rosser has since shown to be inconsistent. Accordingly, in Element and number I dropped the principle *200, in which had been assembled axioms to the effect, roughly, that “stratified functions of elements are elements.” In lieu of *200 I set forth alternatives in which no appeal is made to stratification. The system of Mathematical logic exclusive of *200 carries over as an unchanging framework; and this framework admits, we know, of a simple consistency proof. My concern in the present paper is to draw attention to certain relationships between this framework and earlier theories.

In 1513 the Spanish adventurer Balboa discovered an ocean three times bigger than any other ocean in the world. When he returned to Europe at the end of the year and told the Spaniards, they probably did not fully believe him. Like Balboa, the symbolic logicians have recently discovered a new vast ocean of thought, and like the Europeans of his day, many people are hardly aware of the discovery or its full implications.

The newly discovered ocean, symbolic logic, is, in its broadest sense, a new science which studies through use of efficient symbols the nature and properties of all nonnumerical relations, seeking precise meanings and necessary conclusions. As an applied science, it holds immense promise. For example, it may give us an unambiguous language for political, economic, and social fields, which will conveniently reflect the structure of these fields and make discussion and analysis easy.