Page:Cantortransfinite.djvu/112

Rh [486] An aggregate with the cardinal number $$\mathfrak{a}.\mathfrak{b}$$ may also be made up out of two aggregates $$M$$ and $$N$$ with the cardinal numbers $$\mathfrak{a}$$ and $$\mathfrak{b}$$ according to the following rule: We start from the aggregate $$N$$ and replace in it every element $$n$$ by an aggregate $$M_n\sim M$$; if, then, we collect the elements of all these aggregates $$M$$ to a whole $$S$$, we see that (7) and consequently $\overline\overline{S} = \mathfrak{a}.\mathfrak{b}$. For, if, with any given law of correspondence of the two equivalent aggregates $$M$$ and $$M_n$$, we denote by $$m$$ the element of $$M$$ which corresponds to the element $$m_n$$ of $$M_n$$, we have (8) and thus the aggregates $$S$$ and $$(M.N)$$ can be referred reciprocally and univocally to one another by regarding $$m_n$$ and $$(m, n)$$ as corresponding elements.

From our definitions result readily the theorems:

(9) (10) (11) because:

$(M.N)\sim (N.M)$, $(M.(N.P))\sim ((M.N).P)$, $(M.(N, P))\sim ((M.N), (M.P))$.

Addition and multiplication of powers are subject,