Page:Cantortransfinite.djvu/135

on the former in such a way that $e_{\nu }$ and $f_{\nu }$ are corresponding elements. For $e_{1}$ the lowest element in rank of the first, must, in the process of imaging, be correlated to the lowest element $f_{1}$ of the second, the next after $e_{1}$ in rank $(e_{2})$ to $f_{2}$, the next after $f_{1}$, and so on. [500] Every other bi-univocal correspondence of the two equivalent aggregates $\{e_{\nu }\}$ and $\{f_{\nu }\}$ is not an "imaging" in the sense which we have fixed above for the theory of types.

On the other hand, let us take an ordered aggregate of the form

$\{e_{\nu }\}$,

where $\nu$ represents all positive and negative finite integers, including $0$, and where likewise

$e_{\nu }<e_{\nu +1}$.

This aggregate has no lowest and no highest element in rank. Its type is, by the definition of a sum given in §8,

$^{*}\!\omega +\omega$.

It is similar to itself in an infinity of ways. For let us consider an aggregate of the same type

$\{f_{\nu '}\}$,

where

$f_{\nu '}<f_{\nu '+1}$.

Then the two ordered aggregates can be so imaged on one another that, if we understand by $\nu _{0}'$ a definite one of the numbers $\nu '$, to the element $e_{\nu '}$ of