On Finite Sets and the Peano Postulates

There are two common characterizations of and of infinite sets which may be stated briefly as follows. I. A set S is if it is empty or if there exists a one-to-one correspondence between S and an initial segment of the natural numbers. If S is not finite, it is LI. A set S is infinite if there exists a proper subset of S which can be put into one-to-one correspondence with S. If S is not infinite, it is finite. The equivalence of these two definitions can be demonstrated (see Wilder, Introduction to the Foundations of hiathematics), but of course the properties of the natural iiumbers must be used. It is the purpose of this note to outline how some of the characteristics of sets can be derived directly from the second definition and then how the Peano postulates for natural numbers can be introduced easily as theorems. In what follows, therefore, we will adopt definition II when we use the words finite or infinite. Two sets, A and B, between which there is a one-to-one correspondence will be called equivalent and this equivalence will be denoted by A-B. Terms such as simple order, well-ordering, and common symbols of set theory will be presumed familiar to the reader. In particular, those terms which are not explicitly defined here will have the meaning given in the text mentioned above. Let S be a set simply ordered by a relation of A 'such that A 'A '.. Then A 'uA''A. But A 'uA