Page:Cantortransfinite.djvu/123

we call its cardinal number (§1) "Aleph-zero" and denote it by $\aleph _{0}$ thus we define

That $\aleph _{0}$ is a transfinite number, that is to say, is not equal to any finite number $\mu$, follows from the simple fact that, if to the aggregate $\{\nu \}$ is added a new element $e_{0}$, the union-aggregate $(\{\nu \},e_{0})$ is equivalent to the original aggregate $\{\nu \}$. For we can think of this reciprocally univocal correspondence between them: to the element $e_{0}$ of the first corresponds the element $1$ of the second, and to the element $\nu$ of the first corresponds the element $\nu +1$ of the other. By §3 we thus have

But we showed in §5 that $\mu +1$ is always different from $\mu$ and therefore $\aleph _{0}$ is not equal to any finite number $\mu$.

The number $\aleph _{0}$ is greater than any finite number $\mu$:

[493] This follows, if we pay attention to §3, from the three facts that $\mu ={\overline {\overline {(1,2,3,...,\mu )}}}$, that no part of the aggregate $(1,2,3,...,\mu )$ is equivalent to the aggregate $\{\nu \}$, and that $(1,2,3,...,\mu )$ is itself a part of $\{\nu \}$.

On the other hand, $\aleph _{0}$ is the least transfinite cardinal number. If ${\mathfrak {a}}$ is any transfinite cardinal number different from $\aleph _{0}$, then