Ordinal Addition

Let  and  be disjoint totally ordered sets with order types  and . Then the ordinal sum is defined at set  where, if  and  are both from the same subset, the order is the same as in the subset, but if  is from  and  is from , then  has order type  (Ciesielski 1997, p. 48; Dauben 1990, p. 104; Moore 1982, p. 40).

One should note that in the infinite case, order type addition is not commutative, although it is associative. For example,

(1)

In addition, , with  the least element, is order isomorphic to , but not to , with  the greatest element, since it has a greatest element and the other does not.

An inductive definition for ordinal addition states that for any ordinal number ,

(2)

and

(3)

If  is a limit ordinal, then  is the least ordinal greater than any ordinal in the set  (Rubin 1967, p. 188; Suppes 1972, p. 205).

REFERENCES:

Ciesielski, K. Set Theory for the Working Mathematician. Cambridge, England: Cambridge University Press, 1997.

Dauben, J. W. Georg Cantor: His Mathematics and Philosophy of the Infinite. Princeton, NJ: Princeton University Press, 1990.

Moore, G. H. Zermelo's Axiom of Choice: Its Origin, Development, and Influence. New York: Springer-Verlag, 1982.

Rubin, J. E. Set Theory for the Mathematician. New York: Holden-Day, 1967.

Suppes, P. Axiomatic Set Theory. New York: Dover, 1972.