Page:Cantortransfinite.djvu/138

§8
Addition and Multiplication of Ordinal Types

The union-aggregate $(M,N)$ of two aggregates $M$ and $N$ can, if $M$ and $N$ are ordered, be conceived as an ordered aggregate in which the relations of precedence of the elements of $M$ among themselves as well as the relations of precedence of the elements of $N$ among themselves remain the same as in $M$ or $N$ respectively, and all elements of $M$ have a lower rank than all the elements of $N$. If $M'$ and $N'$ are two other ordered aggregates, $M\simeq M'$ and $N\simeq N'$, [502] then $(M,N)\simeq (M',N')$; so the ordinal type of $(M,N)$ depends only on the ordinal types $M=\alpha$ and $N=\beta$. Thus, we define:

In the sum $\alpha +\beta$ we call $\alpha$ the "augend" and $\beta$ the "addend."

For any three types we easily prove the associative law:

On the other hand, the commutative law is not valid, in general, for the addition of types. We see this by the following simple example.

If $\omega$ is the type, already mentioned in §7, of the well-ordered aggregate

$E=(e_{1},e_{2},...,e_{\nu },...),\quad e_{\nu +1}$,