The Principles of Mathematics (1903)

§ 118

Among finite classes, if one is a proper part of another, the one has a smaller number of terms than the other. (A proper part is a part not the whole.) But among infinite classes, this no longer holds. This distinction is, in fact, an essential part of the above definitions of the finite and the infinite. Of two infinite classes, one may have a greater or a smaller number of terms than the other. A class u is said to be greater than a class v, or to have a number greater than that of v, when the two are not similar, but v is similar to a proper part of u. It is known that if u is similar to a proper part of v, and v to a proper part of u (a case which can only arise when u and v are infinite), then u is similar to v; hence u is greater than v is inconsistent with v is greater than u. It is not at present known whether, of two different infinite numbers, one must be greater and the other less. But it is known that there is a least infinite number, i.e. a number which is less than any different infinite number. This is the number of finite integers, which will be denoted, in the present work, by a0[83]. This number is capable of several definitions in which no mention is made of the finite numbers. In the first place it may be defined (as is implicitly done by Cantor[84]) by means of the principle of mathematical induction. This definition is as follows: a0 is the number of any class u which is the domain of a one-one relation R, whose converse domain is contained in but not coextensive with u, and which is such that, calling the term to which x has the relation R the successor of x, if s be any class to which belongs a term of u which is not a successor of any other term u, and to which belongs the successor of every term of u which belongs to s, then every term of u belongs to s. Or again, we may define a0 as follows. Let P be a transitive and asymmetrical relation, and let any two different terms of the field of P have the relation P or its converse. Further let any class u contained in the field of P and having successors (i.e. terms to which every term of u has the relation P) have an immediate successor, i.e. a term whose predecessors either belong to u or precede some term of u; let there be one term of the field of P which has no predecessors, but let every term which has predecessors have successors and also have an immediate predecessor; then the number of terms in the field of P is a0. Other definitions may be suggested, but as all are equivalent it is not necessary to multiply them. The following characteristic is important: Every class whose number is a0 can be arranged in a series having consecutive terms, a beginning but no end, and such that the number of predecessors of any term of the series is finite; and any series having these characteristics has the number a0.(§ 118 ¶ 1)

It is very easy to show that every infinite class contains classes whose number is a0. For let u be such a class, and let x0 be a term of u. Then u is similar to the class obtained by taking away x, which we will call the class u1. Thus u1 is an infinite class. From this we can take away a term x1, leaving an infinite class u2, and so on. The series of terms x1, x2, … is contained in u, and is of the type which has the number a0. From this point we can advance to an alternative definition of the finite and the infinite by means of mathematical induction, which must now be explained.(§ 118 ¶ 2)

§ 118 n. 1. Cantor employs for this number the Hebrew Aleph with the suffix 0, but this notation is inconvenient.

§ 118 n. 2. Math. Annalen, Vol. XLVI, § 6.