Page:Cantortransfinite.djvu/139

then ${\displaystyle 1+\omega }$ is not equal to ${\displaystyle \omega +1}$. For, if ${\displaystyle f}$ is a new element, we have by (1):

But the aggregate

is similar to the aggregate E, and consequently

On the contrary, the aggregates ${\displaystyle E}$ and ${\displaystyle (E,f)}$ are not similar, because the first has no term which is highest in rank, but the second has the highest term ${\displaystyle f}$. Thus ${\displaystyle \omega +1}$ is different from ${\displaystyle \omega =1+\omega }$.

Out of two ordered aggregates ${\displaystyle M}$ and ${\displaystyle N}$ with the types ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ we can set up an ordered aggregate ${\displaystyle S}$ by substituting for every element ${\displaystyle n}$ of ${\displaystyle N}$ an ordered aggregate ${\displaystyle M_{n}}$ which has the same type ${\displaystyle \alpha }$ as ${\displaystyle M}$, so that

(3)

${\displaystyle {\overline {M}}_{n}=\alpha }$;

and, for the order of precedence in

(4)

${\displaystyle S=\{M_{n}\}}$

we make the two rules:

(1) Every two elements of ${\displaystyle S}$ which belong to one and the same aggregate ${\displaystyle M_{n}}$ are to retain in ${\displaystyle S}$ the same order of precedence as in ${\displaystyle M_{n}}$;

(2) Every two elements of ${\displaystyle S}$ which belong to two different aggregates ${\displaystyle M_{n_{1}}}$ and ${\displaystyle M_{n_{2}}}$ have the same relation of precedence as ${\displaystyle n_{1}}$ and ${\displaystyle n_{2}}$ have in ${\displaystyle N}$.