Page:Cantortransfinite.djvu/116

and thus (§1)

$2^{\aleph _{0}}={\overline {\overline {X}}}={\mathfrak {o}}$.

From (11) follows by squaring (by § 6, (6))

${\mathfrak {o}}\cdot {\mathfrak {o}}=2^{\aleph _{0}}\cdot 2^{\aleph _{0}}=2^{\aleph _{0}+\aleph _{0}}={\mathfrak {o}}$,

and hence, by continued multiplication by ${\mathfrak {o}}$,

where $\nu$ is any finite cardinal number.

If we raise both sides of (11) to the power^[1] $\aleph _{0}$ we get

${\mathfrak {o}}^{\aleph _{0}}=(2^{\aleph _{0}})^{\aleph _{0}}=2^{\aleph _{0}\cdot \aleph _{0}}$.

But since, by § 6, (8), $\aleph _{0}\cdot \aleph _{0}=\aleph _{0}$, we have

The formulae (13) and (14) mean that both the $\nu$-dimensional and the $\aleph _{0}$-dimensional continuum have the power of the one-dimensional continuum. Thus the whole contents of my paper in Crelle's Journal, vol. lxxxiv, 1878,^[2] are derived purely algebraically with these few strokes of the pen from the fundamental formulæ of the calculation with cardinal numbers.

The Finite Cardinal Numbers

We will next show how the principles which we have laid down, and on which later on the theory of the actually infinite or transfinite cardinal numbers

1. [In English there is an ambiguity.]
2. [See Section V of the Introduction.]

7