Page:Cantortransfinite.djvu/112

[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

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

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:

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,