Page:Cantortransfinite.djvu/139

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

But the aggregate

is similar to the aggregate E, and consequently

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

Out of two ordered aggregates $$M$$ and $$N$$ with the types $$\alpha$$ and $$\beta$$ we can set up an ordered aggregate $$S$$ by substituting for every element $$n$$ of $$N$$ an ordered aggregate $$M_n$$ which has the same type $$\alpha$$ as $$M$$, so that (3) and, for the order of precedence in (4) we make the two rules:

(1) Every two elements of $$S$$ which belong to one and the same aggregate $$M_n$$ are to retain in $$S$$ the same order of precedence as in $$M_n$$;

(2) Every two elements of $$S$$ which belong to two different aggregates $$M_{n_1}$$ and $$M_{n_2}$$ have the same relation of precedence as $$n_1$$ and $$n_2$$ have in $$N$$.