On Series of Ordinals and Combinatorics

✍ Scribed by James P. Jones; Hilbert Levitz; Warren D. Nichols

• Publisher: John Wiley and Sons
• Year: 1997
• Tongue: English
• Weight: 660 KB
• Volume: 43
• Category: Article
• ISSN: 0044-3050
• DOI: 10.1002/malq.19970430114

✦ Synopsis

Abstract

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.