Page:Cantortransfinite.djvu/123

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

(1)

${\displaystyle \aleph _{0}={\overline {\overline {\{\nu \}}}}}$.

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

(2)

${\displaystyle \aleph _{0}+1=\aleph _{0}}$

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

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

(3)

${\displaystyle \aleph _{0}>\mu }$

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

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

(4)

${\displaystyle \aleph _{0}<{\mathfrak {a}}}$.