Page:Cantortransfinite.djvu/140

The ordinal type of $S$ depends, as we easily see, only on the types $\alpha$ and $\beta$; we define

[503] In this product $\alpha$ is called the "multiplicand" and $\beta$ the "multiplier."

In any definite imaging of $M$ on $M_{n}$ let $m_{n}$ be the element of $M_{n}$ that corresponds to the element $m$ of $M$; we can then also write

Consider a third ordered aggregate $P=\{p\}$ with

$\alpha \,.\beta ={\overline {\{m_{n}\}}},\quad \beta \,.\gamma ={\overline {\{n_{p}\}}},\quad (\alpha \,.\beta )\,.\gamma ={\overline {\{(m_{n})_{p}\}}},\quad \alpha \,.(\beta \,.\gamma )={\overline {\{m_{(n_{p})}\}}}$.

But the two ordered aggregates $\{(m_{n})_{p}\}$ and $\{m_{(n_{p})}\}$ are similar, and are imaged on one another if we regard the elements $(m_{n})_{p}$ and $m_{(n_{p})}$ as corresponding.

Consequently, for three types $\alpha$, $\beta$, and $\gamma$ the associative law

subsists. From (1) and (5) follows easily the distributive law

but only in this form, where the factor with two terms is the multiplier.

On the contrary, in the multiplication of types as in their addition, the commutative law is not