Page:Cantortransfinite.djvu/111

Since in the conception of power, we abstract from the order of the elements, we conclude at once that

and, for any three cardinal numbers ${\mathfrak {a}}$, ${\mathfrak {b}}$, ${\mathfrak {c}}$, we have

We now come to multiplication. Any element $m$ of an aggregate $M$ can be thought to be bound up with any element $n$ of another aggregate $N$ so as to form a new element $(m,n)$; we denote by $(M.N)$ the aggregate of all these bindings $(m,n)$, and call it the "aggregate of bindings (Verbindungsmenge) of $M$ and $N$." Thus

We see that the power of $(M.N)$ only depends on the powers ${\overline {\overline {M}}}={\mathfrak {a}}$ and ${\overline {\overline {N}}}={\mathfrak {b}}$; for, if we replace the aggregates $M$ and $N$ by the aggregates

$M'={m'}$ and $N'={n'}$

respectively equivalent to them, and consider $m$, $m'$ and $n$, $n'$ as corresponding elements, then the aggregate

$(M'.N')={(m',n')}$

is brought into a reciprocal and univocal correspondence with $(M.N)$ by regarding $(m,n)$ and $(m',n')$ as corresponding elements. Thus

We now define the product ${\mathfrak {a}}.{\mathfrak {b}}$ by the equation