Page:Cantortransfinite.djvu/117

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

Let us now unite with $E_{0}$ another thing $e_{1}$ and call the union-aggregate $e_{1}$, so that

The cardinal number of $E_{1}$ is called "two" and is denoted by 2:

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,