All Numbers Great and Small

Starting with the ordered set Q of rational numbers, Dedekind defined a real number to be an ordered pair (L, R), where L < R, L ∪ R= Q and L, R ≠o. von Neumann later identified each ordinal with the set of all ordinals previously created, so that, 0 = o, 1 = {0}, … , ω = {0, 1, …}, and so on. More recently, Conway [5, 6] discovered that these two methods can be subsumed under a more general construction which leads to an ordered class of numbers embracing the reals and the ordinals as well as many less familiar numbers including -ω, ω/2, 1/ω, (Math) and ω – πt to name only a few. He further showed that the arithmetic of the reals may be extended to the entire class yielding a real-closed ordered field.