Page:Cantortransfinite.djvu/138

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

(1)

${\displaystyle \alpha +\beta =({\overline {M,N}})}$.

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

For any three types we easily prove the associative law:

(2)

${\displaystyle \alpha +(\beta +\gamma )=(\alpha +\beta )+\gamma }$.

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