Page:Cantortransfinite.djvu/142

Rh determines its rank. We denote the ordinal type of $$R$$ by $$\eta$$: (1) 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) (3) 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) 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: