Page:Cantortransfinite.djvu/142

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OF TRANSFINITE NUMBERS

123

determines its rank. We denote the ordinal type of $R$ by $\eta$ :

(1)

\eta ={\overline {R}}

.

But we have put the same aggregate in another order of precedence in which we call it $R_{0}$ . This order is determined, in the first place, by the magnitude of $p+q$ , and in the second place—for rational numbers for which $p+q$ has the same value—by the magnitude of $p/q$ itself. The aggregate $R_{0}$ is a well-ordered aggregate of type $\omega$ :

(2)

R_{0}=(r_{1},r_{2},...,r_{\nu },...)

, where

r_{\nu }<r_{\nu +1}

,

(3)

{\overline {R_{0}}}=\omega

.

Both $R$ and $R_{0}$ have the same cardinal number since they only differ in the order of precedence of their elements, and, since we obviously have ${\overline {\overline {R_{0}}}}=\aleph _{0}$ , we also have

(4)

{\overline {\overline {R}}}={\overline {\eta }}=\aleph _{0}

.

Thus the type $\eta$ belongs to the class of types $[\aleph _{0}]$ .

Secondly, we remark that in $R$ there is neither an element which is lowest in rank nor one which is highest in rank. Thirdly, $R$ has the property that between every two of its elements others lie. This property we express by the words: $R$ is "everywhere dense" (überalldicht).

We will now show that these three properties characterize the type $\eta$ of $R$ , so that we have the following theorem:

Page:Cantortransfinite.djvu/142

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