Page:Cantortransfinite.djvu/124

This rests on the following theorems:

A. Every transfinite aggregate $T$ has parts with the cardinal number $\aleph _{0}$.

Proof.—If, by any rule, we have taken away a finite number of elements $t_{1},t_{2},t_{3},...,t_{\nu -1}$, there always remains the possibility of taking away a further element $t_{\nu }$. The aggregate $\{t_{\nu }\}$, where $\nu$ denotes any finite cardinal number, is a part of $T$ with the cardinal number $\aleph _{0}$, because $\{t_{\nu }\}\sim \{\nu \}$ (§1).

B. If $S$ is a transfinite aggregate with the cardinal number $\aleph _{0}$, and $S_{1}$ is any transfinite part of $S$, then ${\overline {\overline {S_{1}}}}=\aleph _{0}$.

Proof.—We have supposed that $S\sim \{\nu \}$. Choose a definite law of correspondence between these two aggregates, and, with this law, denote by $s_{\nu }$ that element of $S$ which corresponds to the element $\nu$ of $\{\nu \}$, so that

$S=\{s_{\nu }\}$

The part $S_{1}$ of $S$ consists of certain elements $s_{\kappa }$ of $S$, and the totality of numbers $\kappa$ forms a transfinite part $K$ of the aggregate $\{\nu \}$. By theorem G of §5 the aggregate $K$ can be brought into the form of a series

$K=\{\kappa _{\nu }\}$

where

$\kappa _{\nu }<\kappa _{\nu +1}$;

consequently we have

$S_{1}=\{s_{\kappa _{\nu }}\}$.