Transfinite numbers are infinite ordinal numbers. Informally, ordinal numbers may be represented by strings of *'s. For example, the ordinal number 0 is the string of no stars. Number 1 is *. Number 2 is ** and so on. The first transfinite number is then represented by ***.... Here the three dots after the *** imply that the sequence repeats infinitely. This number is traditionally called omega and is written . Ordinal numbers are added by juxtaposition. For example 1 + 2 = (*) + (**) = *** = 3. So 1 + = ****... = . But + 1 = ****...*. These two numbers are different for the following reason. If a frog is on the first star of ***..., and it can leap onto successive stars, then it can reach any star in ***.... But, a frog on the first star of ***...* cannot get to the last star by hopping consecutive stars. So, addition of transfinite numbers is not commutative.

Suppose that x and y are two transfinite numbers. Then x times y is represented...