¹ This theory is the outcome of work done by the author while holding a John Simon Guggenheim Memorial Foundation Fellowship during the academic year 1945–46. It is hoped eventually to make use of it in connection with the author's system of “basic logic,” but the main features of the theory are most easily presented in connection with the Russell-Whitehead logic. This theory has a close connection with the theory of positive integers developed by Alonzo Church, J. B. Rosser, and S. C. Kleene. (See, for example, Chapter III of Church's book, The calculi of lambda-conversion, Princeton, 1941.)

² In order to assert that a class has R members, where R is a natural number in the sense described, we simply assert that the class in question bears to the empty class the Rth power of the relation, “has one more member than.” The latter relation can be defined as holding from one class to another when the two classes are different and when the former class is the logical sum of the latter class with some unit class.

³ For example, we could first define (non-negative) real numbers in the classical way as classes of (already defined) non-negative rational numbers. Then we could restrict Μ to relations of the following form between a and b: a is greater than (or less than) b by an amount c, where a, b, and c would be these real numbers. This would make possible a definition of real numbers in a second sense, according to which real numbers would be treated as being exponents attaching to the relations that are the members of Μ. Real numbers in this second sense would then be analogous to the natural numbers and rational numbers of this theory, and they would include such numbers among themselves. Thus the real number 2 would be identical with the natural number 2 for an Μ thus restricted. Negative real numbers would also be available, since the converses of members of Μ are also understood as admitted to membership in Μ in such a scheme.