Page:Cantortransfinite.djvu/115

[488] We see how pregnant and far-reaching these simple formulæ extended to powers are by the following example. If we denote the power of the linear continuum $X$ (that is, the totality $X$ of real numbers $x$ such that $x\geq$ and $\leq 1$) by ${\mathfrak {o}}$, we easily see that it may be represented by, amongst others, the formula:

where § 6 gives the meaning of $\aleph _{0}$. In fact, by (4), $2^{\aleph _{0}}$ is the power of all representations

(where $f(\nu )=0$ or $1$)

of the numbers $x$ in the binary system. If we pay attention to the fact that every number $x$ is only represented once, with the exception of the numbers $x={\frac {2\nu +1}{2^{\mu }}}<1$, which are represented twice over, we have, if we denote the "enumerable" totality of the latter by ${s_{\nu }}$,

$2^{\aleph _{0}}=({\overline {\overline {\{s_{\nu }\},X}}})$.

If we take away from $X$ any "enumerable" aggregate ${t_{\nu }}$ and denote the remainder by $X_{1}$, we have:

$X=(\{t_{\nu }\},X_{1})=(\{t_{2\nu -1}\},\{t_{2\nu }\},X_{1})$,

$(\{s_{\nu }\},X)=(\{s_{\nu }\},\{t_{\nu }\},X_{1})$,

$\{t_{2\nu -1}\}\sim \{s_{\nu }\},\quad \{t_{2\nu }\}\sim \{t_{\nu }\},\quad X_{1}\sim X_{1}$;

so

$X\sim (\{s_{\nu }\},X)$,