Page:Cantortransfinite.djvu/127

108 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 $$\nu$$: (10) 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