Introduction to Mathematical Philosophy

Book may have numerous typos, missing text, images, or index. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. 1920. Excerpt: ... CHAPTER XII SELECTIONS AND THE MULTIPLICATIVE AXIOM In this chapter we have to consider an axiom which can be enunciated, but not proved, in terms of logic, and which is convenient, though not indispensable, in certain portions of mathematics. It is convenient, in the sense that many interesting propositions, which it seems natural to suppose true, cannot be proved without its help; but it is not indispensable, because even without those propositions the subjects in which they occur still exist, though in a somewhat mutilated form. Before enunciating the multiplicative axiom, we must first explain the theory of selections, and the definition of multiplication when the number of factors may be infinite. In defining the arithmetical operations, the only correct procedure is to construct an actual class (or relation, in the case of relation-numbers) having the required number of terms. This sometimes demands a certain amount of ingenuity, but it is essential in order to prove the existence of the number defined. Take, as the simplest example, the case of addition. Suppose we are given a cardinal number fi, and a class a which has fi terms. How shall we define j^+ju.? For this purpose we must have two classes having /x terms, and they must not overlap. We can construct such classes from a in various ways, of which the following is perhaps the simplest: Form first all the ordered couples whose first term is a class consisting of a single member of a, and whose second term is the null-class; then, secondly, form all the ordered couples whose first term is the null-class and whose second term is a class consisting of a single member of a. These two classes of couples have no member in common, and the logical sum of the two classes will have terms. Exactly analogously we can define fi--v, g...