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On Series of Ordinals and Combinatorics
Author(s) -
Jones James P.,
Levitz Hilbert,
Nichols Warren D.
Publication year - 1997
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.19970430114
Subject(s) - mathematics , finitary , natural number , combinatorics , element (criminal law) , series (stratigraphy) , order type , class (philosophy) , discrete mathematics , set (abstract data type) , partially ordered set , function (biology) , order (exchange) , cover (algebra) , limit (mathematics) , infinite set , finite set , mathematical analysis , mechanical engineering , paleontology , finance , artificial intelligence , evolutionary biology , political science , computer science , law , economics , biology , programming language , engineering
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.