On Series of Ordinals and Combinatorics

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