The Principles of Mathematics (1903)

§ 117

The purpose of the present chapter is not to discuss the philosophical difficulties concerning the infinite, which are postponed to Part V. For the present I wish merely to set forth briefly the mathematical theory of finite and infinite as it appears in the theory of cardinal numbers. This is its most fundamental form, and must be understood before the ordinal infinite can be adequately explained[82].(§ 117 ¶ 1)

Let u be any class, and let u′ be a class formed by taking away one term x from u. Then it may or may not happen that u is similar to u′. For example, if u be the class of all finite numbers, and u′ the class of all finite numbers except 0, the terms of u′ are obtained by adding 1 to each of the terms of u, and this correlates one term of u with one of u′ and vice versâ, no term of either being omitted or taken twice over. Thus u′ is similar to u. If there is one term x which can be taken away from u to leave a similar class u′, it is easily proved that if any other term y is taken away instead of x we also get a class similar to u. When it is possible to take away one term from u and leave a class u′ similar to u, we say that u is an infinite class. When this is not possible, we say that u is a finite class. From these definitions it follows that the null-class is finite, since no term can be taken from it. It is also easy to prove that if u be a finite class, the class formed by adding one term to u is finite; and conversely if this class is finite, so is u. It follows from the definition that the numbers of finite classes other than the null-class are altered by subtracting 1, while those of the infinite classes are unaltered by this operation. It is easy to prove that the same holds of the addition of 1.(§ 117 ¶ 2)

§ 117 n. 1. On the present topic cf. Cantor, Math. Annalen, Vol. XLVI, §§ 5, 6, where most of what follows will be found.