{"history":[],"article":{"id":"7756e6a3-3b70-47d7-8c2a-d1b30eca7a89","title":"PCF theory","slug":"pcf-theory","content":":::summary\n:::\n\n# \n**PCF theory** is the name of a [mathematical](https://bliptext.com/articles/mathematics) theory, introduced by Saharon, that deals with the [cofinality](https://bliptext.com/articles/cofinality) of the [ultraproducts](https://bliptext.com/articles/ultraproduct) of [ordered sets](https://bliptext.com/articles/ordered-set).\nIt gives strong upper bounds on the cardinalities of [power sets](https://bliptext.com/articles/power-set) of [singular](https://bliptext.com/articles/regular-cardinal) [cardinals](https://bliptext.com/articles/cardinal-number), and has many more applications as well.\nThe abbreviation \"PCF\" stands for \"possible [cofinalities](https://bliptext.com/articles/cofinality)\".\n\n# Main definitions\nIf A is an infinite set of [regular cardinals](https://bliptext.com/articles/regular-cardinal), D is an [ultrafilter](https://bliptext.com/articles/ultrafilter) on A, then\nwe let denote the cofinality of the ordered set of functions\n\\prod A where the ordering is defined as follows:\nf

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