Page:Cantortransfinite.djvu/117

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98
THE FOUNDING OF THE THEORY

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

To a single thing , if we subsume it under the concept of an aggregate , corresponds as cardinal number what we call "one" and denote by 1; we have

(1)
.

Let us now unite with another thing and call the union-aggregate , so that

(2)
.

The cardinal number of is called "two" and is denoted by 2:

(3)
.

By addition of new elements we get the series of aggregates

,

which give us successively, in unlimited sequence, the other so-called "finite cardinal numbers" denoted by , , , ... 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 is understood the number immediately preceding in the above series,

(4)
,
(5)
.