Page:Cantortransfinite.djvu/138

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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: (1) In the sum $$\alpha+\beta$$ we call $$\alpha$$ the "augend" and $$\beta$$ the "addend."

For any three types we easily prove the associative law: (2) 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