Page:Cantortransfinite.djvu/111

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

(2)

${\displaystyle {\mathfrak {a}}+{\mathfrak {b}}={\mathfrak {b}}+{\mathfrak {a}}}$;

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

(3)

${\displaystyle {\mathfrak {a}}+({\mathfrak {b}}+{\mathfrak {c}})=({\mathfrak {a}}+{\mathfrak {b}})+{\mathfrak {c}}}$.

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

(4)

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

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

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

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

(5)

${\displaystyle (M'.N')\sim (M.N)}$.

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

(6)

${\displaystyle {\mathfrak {a}}.{\mathfrak {b}}=({\overline {\overline {M.N}}})}$.