Page:Cantortransfinite.djvu/141

generally valid. For example, $2\,.\omega$ and $\omega \,.2$ are different types; for, by (5),

$2\,.\omega ={\overline {(e_{1},f_{1};e_{2},f_{2};...;e_{\nu },f_{\nu };...)}}=\omega$;

while

$\omega \,.2={\overline {(e_{1},e_{2},...,e_{\nu },...;f_{1},f_{2},...,f_{\nu },...)}}$

is obviously different from $\omega$.

If we compare the definitions of the elementary operations for cardinal numbers, given in §3, with those established here for ordinal types, we easily see that the cardinal number of the sum of two types is equal to the sum of the cardinal numbers of the single types, and that the cardinal number of the product of two types is equal to the product of the cardinal numbers of the single types. Every equation between ordinal types which proceeds from the two elementary operations remains correct, therefore, if we replace in it all the types by their cardinal numbers.

The Ordinal Type $\eta$ of the Aggregate $R$ of all Rational Numbers which are Greater than $0$ and Smaller than $1$, in their Natural Order of Precedence

By $R$ we understand, as in §7, the system of all rational numbers $p/q$ ($p$ and $q$ being relatively prime) which $>0$ and $<1$, in their natural order of precedence, where the magnitude of a number