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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 $\nu$;