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.)

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.

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.