Ordinal Multiplication

Let  and  be totally ordered sets. Let  be the Cartesian product and define order as follows. For any  and ,

1. If , then ,

2. If , then  and  compare the same way as  (i.e., lexicographical order)

(Ciesielski 1997, p. 48; Rubin 1967; Suppes 1972). However, Dauben (1990, p. 104) and Moore (1982, p. 40) define multiplication in the reverse order.

Like addition, multiplication is not commutative, but it is associative,

(1)

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

(2)

(3)

If  is a limit ordinal, then  is the least ordinal greater than any ordinal in the set  (Suppes 1972, p. 212).

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.