Cardinal number
Cardinal number
Two classes are equivalent if there exists a one-to-one correspondence between them (see One-one). Cardinal numbers are obtained by abstraction (q. v.) with respect to equivalence, so that two classes have the same cardinal number if and only if they are equivalent. This may be formulated more exactly, following Frege, by defining the cardinal number of a class to be the class of classes equivalent to it.
If two classes a and b have no members in common, the cardinal number of the logical sum of a and b is uniquely determined by the cardinal numbers of a and b, and is called the sum of the cardinal number of a and the cardinal number of b.
0 is the cardinal number of the null class. 1 is the cardinal number of a unit class (all unit classes have the same cardinal number).
A cardinal number is inductive if it is a member of every class t of cardinal numbers which has the two properties, (1) 0? t, and (2) for all x, if x? t and y is the sum of x and 1, then y? t. In other (less exact) words, the inductive cardinal numbers are those which can be reached from 0 by successive additions of 1. A class b is infinite if there is a class a, different from b, such that a ? b and a is equivalent to b. In the contrary case b is finite. The cardinal number of an infinite class is said to be infinite, and of a finite class, finite. It can be proved that every inductive cardinal number is finite, and, with the aid of the axiom of choice, that every finite cardinal number is inductive.
The most important infinite cardinal number is the cardinal number of the class of inductive cardinal numbers (0, 1, 2, . . .); it is called aleph-zero and symbolized by a Hebrew letter aleph followed by an inferior 0.
For brevity and simplicity in the preceding account we have ignored complications introduced by the theory of types, which are considerable and troublesome. Modifications are also required if the account is to be incorporated into the Zermelo set theory. –A.C. G. Cantor, Contributions to the Founding of the Theory of Trasfinite Numbers, translated and with an introduction bv P.E.B. Jourdain, Chicago and London, 1915. Whitehead and Russell, Principia Mathematica, vol. 2.