Page:Cantortransfinite.djvu/123

104 we call its cardinal number (§1) "Aleph-zero" and denote it by $$\aleph_0$$ thus we define (1) 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 (2) 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 $$\mu$$: (3) [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 (4)