On Series of Ordinals and Combinatorics

This paper deals mainly with generalizations of results in nitary combinatorics to innnite ordinals. It is well-known that for nite ordinals P << is the number of 2-element subsets of an-element set. It is shown here that for any well-ordered set of arbitrary innnite order type , P << is the ordinal of the set M of 2-element subsets where M is ordered in some natural way. This result is then extended to evaluating the ordinal of the set of all n-element subsets for each natural number n 2 Moreover series P << f() are investigated and evaluated, where is a limit ordinal and the function f belongs to a certain class of functions containing polynomials with natural number coeecients. The tools developed for this result can be extended to cover all innnite , but the case of nite appears to be quite problematic.