Page:Cantortransfinite.djvu/124

This rests on the following theorems:

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

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

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

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

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

where

consequently we have