ABSTRACT
The mathematical theory of infinity may almost be said to begin with Cantor. The transfinite cardinals may be defined in the first place so as to include the finite cardinals, leaving it to be investigated in what respects the finite and the transfinite are distinguished. This chapter examines the chief properties of cardinal numbers. The difference between finite and transfinite cardinals results from the defining difference of finite and infinite, namely that when the number of a class is not finite, it always has a proper part which is similar to the whole. The proposition itself may be taken as the definition of the transfinite among cardinal numbers, for it is a property belonging to all of them, and to none of the finite cardinals. Among transfinite cardinals, some are particularly important, especially the number of finite numbers, and the number of the continuum.
