Page:Cantortransfinite.djvu/134

It may happen that $^{*}\!\alpha =\alpha$, as, for example, in the case of finite types or in that of the type of the aggregate of all rational numbers which are greater than $0$ and less than $1$ in their natural order of precedence. This type we will investigate under the notation $\eta$.

We remark further that two similarly ordered aggregates can be imaged on one another either in. one manner or in many manners; in the first case the type in question is similar to itself in only one way, in the second case in many ways. Not only all finite types, but the types of transfinite "well-ordered aggregates," which will occupy us later and which we call transfinite "ordinal numbers," are such that they allow only a single imaging on themselves. On the other hand, the type $\eta$ is similar to itself in an infinity of ways.

We will make this difference clear by two simple examples. By $\omega$ we understand the type of a well-ordered aggregate

$(e_{1},e_{2},...,e_{\nu },...)$,

in which

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

and where $\nu$ represents all finite cardinal numbers in turn. Another well-ordered aggregate

$(f_{1},f_{2},...,f_{\nu },...)$,

with the condition

$f_{\nu }<f_{\nu +1}$,

of the same type $\omega$ can obviously only be imaged