Page:Cantortransfinite.djvu/122

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OF TRANSFINITE NUMBERS
103

the number , in which case it is the least, , or it does not. In the latter case, let be the aggregate of all those cardinal numbers of our series, , which are smaller than^those occurring in . If a number belongs to , all numbers less than belong to . But must have one element such that , and consequently all greater numbers, do not belong to , because otherwise would contain all finite numbers, whereas the numbers belonging to are not contained in . Thus is the segment (Abschnitt) (). The number is necessarily an element of and smaller than the rest.

From F we conclude:

G. Every aggregate of different finite cardinal numbers can be brought into the form of a series

such that

§6
The Smallest Transfinite Cardinal Number Aleph-Zero

Aggregates with finite cardinal numbers are called "finite aggregates," all others we will call "transfinite aggregates" and their cardinal numbers "transfinite cardinal numbers."

The first example of a transfinite aggregate is given by the totality of finite cardinal numbers ;