Singular Cardinals and the PCF Theory

E-ISSN： 1943-5894|1|4|408-424

ISSN： 1079-8986

Source： Bulletin of Symbolic Logic, Vol.1, Iss.4, 1995-12, pp. : 408-424

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Abstract

§1. Introduction. Among the most remarkable discoveries in set theory in the last quarter century is the rich structure of the arithmetic of singular cardinals, and its deep relationship to large cardinals. The problem of finding a complete set of rules describing the behavior of the continuum function 2^ℵ_α for singular ℵ_α's, known as the Singular Cardinals Problem, has been attacked by many different techniques, involving forcing, large cardinals, inner models, and various combinatorial methods. The work on the singular cardinals problem has led to many often surprising results, culminating in a beautiful theory of Saharon Shelah called the pcf theory (“pcf” stands for “possible cofinalities”). The most striking result to date states that if 2^ℵ_n < ℵ_ω for every n = 0, 1, 2, …, then 2^ℵ_ω < ℵ_ω4.