Page:Cantortransfinite.djvu/117

98 will be built, afford also the most natural, shortest, and most rigorous foundation for the theory of finite numbers.

To a single thing $$e_0$$, if we subsume it under the concept of an aggregate $$E_0 = (e_o)$$, corresponds as cardinal number what we call "one" and denote by 1; we have (1) Let us now unite with $$E_0$$ another thing $$e_1$$ and call the union-aggregate $$e_1$$, so that (2) The cardinal number of $$E_1$$ is called "two" and is denoted by 2: (3) By addition of new elements we get the series of aggregates $E_2 = (E_1, e_2), \quad E_3 = (E_2, e_3), ...$, which give us successively, in unlimited sequence, the other so-called "finite cardinal numbers" denoted by $$3$$, $$4$$, $$5$$, ... The use which we here make of these numbers as suffixes is justified by the fact that a number is only used as a suffix when it has been defined as a cardinal number. We have, if by $$\nu-1$$ is understood the number immediately preceding $$\nu$$ in the above series, (4) (5)