Introduction to Mathematical Philosophy

By Bertrand Russell | Go to book overview

CHAPTER XIII
THE AXIOM OF INFINITY AND LOGICAL TYPES

THE axiom of infinity is an assumption which may be enunciated as follows:--

"If n be any inductive cardinal number, there is at least one class of individuals having n terms."

If this is true, it follows, of course, that there are many classes of individuals having n terms, and that the total number of individuals in the world is not an inductive number. For, by the axiom, there is at least one claw having n+1 terms, from which it follows that there are many classes of n terms and that n is not the number of individuals in the world. Since n is any inductive number, it follows that the number of individuals in the world must (if our axiom be true) exceed any inductive number. In view of what we found in the preceding chapter, about the possibility of cardinals which are neither inductive nor reflexive, we cannot infer from our axiom that there are at least א0 individuals, unless we assume the multiplicative axiom. But we do know that there are at least א0 classes of classes, since the inductive cardinals are classes of classes, and form a progression if our axiom is true. The way in which the need for this axiom arises may be explained as follows:--One of Peano's assumptions is that no two inductive cardinals have the same successor, i.e. that we shall not have m+1=n+1 unless m=n, if m and n are inductive cardinals. In Chapter VIII. we had occasion to use what is virtually the same as the above assumption of Peano's, namely, that, if n is an inductive cardinal,

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