Page:Cantortransfinite.djvu/115

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96

THE FOUNDING OF THE THEORY

[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:

(11)

{\mathfrak {o}}=2^{\aleph _{0}}

,

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

(12)

x={\frac {f(1)}{2}}+{\frac {f(2)}{2^{2}}}+...+{\frac {f(\nu )}{2^{\nu }}}+...

(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

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