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).

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