Page:Cantortransfinite.djvu/127

reciprocally univocal relation subsists between the aggregates $\{\nu \}$ and $\{(\mu ,\nu )\}$.

[495] If both sides of the equation (8) are multiplied by $\aleph _{0}$, we get $\aleph _{0}^{3}=\aleph _{0}^{2}=\aleph _{0}$, and, by repeated multiplications by $\aleph _{0}$, we get the equation, valid for every finite cardinal number $\nu$:

The theorems E and A of §5 lead to this theorem on finite aggregates:

C. Every finite aggregate $E$ is such that it is equivalent to none of its parts.

This theorem stands sharply opposed to the following one for transfinite aggregates:

D. Every transfinite aggregate $T$ is such that it has parts $T_{1}$ which are equivalent to it.

Proof.—By theorem A of this paragraph there is a part $S=\{t_{\nu }\}$ of $T$ with the cardinal number $\aleph _{0}$. Let $T=(S,U)$, so that $U$ is composed of those elements of $T$ which are different from the elements $t_{\nu }$. Let us put $S_{1}=\{t_{\nu +1}\}$, $T_{1}=(S-1,U)$; then $T_{1}$ is a part of $T$, and, in fact, that part which arises out of $T$ if we leave out the single element $t_{1}$. Since $S\sim S_{1}$, by theorem B of this paragraph, and $U\sim U$, we have, by §1, $T\sim T_{1}$.

In these theorems C and D the essential difference between finite and transfinite aggregates, to which I referred in the year 1877, in volume lxxxiv [1878] of Crelle's Journal, p. 242, appears in the clearest way.

After we have introduced the least transfinite