PCF theory

PCF theory is the name of a mathematical theory, introduced by Saharon Shelah (1978), that deals with the cofinality of the ultraproducts of ordered sets. It gives strong upper bounds on the cardinalities of power sets of singular cardinals, and has many more applications as well. The abbreviation "PCF" stands for "possible cofinalities".

If A is an infinite set of regular cardinals, D is an ultrafilter on A, then we let ${\displaystyle cf(\prod A/D)}$ denote the cofinality of the ordered set of functions ${\displaystyle \prod A}$ where the ordering is defined as follows. ${\displaystyle f<g}$ if ${\displaystyle \{x\in A:f(x)<g(x)\}\in D}$. pcf(A) is the set of cofinalities that occur if we consider all ultrafilters on A, that is,

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