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Rh the number $$1$$, in which case it is the least, $$\kappa_1 = 1$$, or it does not. In the latter case, let $$J$$ be the aggregate of all those cardinal numbers of our series, $$1, 2, 3, ...$$, which are smaller than^those occurring in $$K$$. If a number $$\nu$$ belongs to $$J$$, all numbers less than $$\nu$$ belong to $$J$$. But $$J$$ must have one element $$\nu_1$$ such that $$\nu_1+1$$, and consequently all greater numbers, do not belong to $$J$$, because otherwise $$J$$ would contain all finite numbers, whereas the numbers belonging to $$K$$ are not contained in $$J$$. Thus $$J$$ is the segment (Abschnitt) ($$1, 2, 3, ..., \nu_1$$). The number $$\nu_1+1 = \kappa_1$$ is necessarily an element of $$K$$ 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 $K=(\kappa_1, \kappa_2, \kappa_3, ...)$ such that $\kappa_1 < \kappa_2 < \kappa_3, ...$

§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 $$\nu$$;