Page:Cantortransfinite.djvu/111

92 Since in the conception of power, we abstract from the order of the elements, we conclude at once that (2) and, for any three cardinal numbers $$\mathfrak{a}$$, $$\mathfrak{b}$$, $$\mathfrak{c}$$, we have (3)

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 (4) 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 (5) We now define the product $$\mathfrak{a}.\mathfrak{b}$$ by the equation (6)