Page:Cantortransfinite.djvu/135

116 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

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

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

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

where

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