Page:Cantortransfinite.djvu/115

96 [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 $$x\ge$$ and $$\le 1$$) by $$\mathfrak{o}$$, we easily see that it may be represented by, amongst others, the formula: (11) where § 6 gives the meaning of $$\aleph_0$$. In fact, by (4), $$2^{\aleph_0}$$ is the power of all representations (12) (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 $X\sim (\{s_\nu\}, X)$,