PCF theory is the name of a
mathematical theory, introduced by Saharon , that deals with the
cofinality
In mathematics, especially in order theory, the cofinality cf(''A'') of a partially ordered set ''A'' is the least of the cardinalities of the cofinal subsets of ''A''.
This definition of cofinality relies on the axiom of choice, as it uses t ...
of the
ultraproducts of
ordered sets. It gives strong upper bounds on the cardinalities of
power sets of
singular
Singular may refer to:
* Singular, the grammatical number that denotes a unit quantity, as opposed to the plural and other forms
* Singular homology
* SINGULAR, an open source Computer Algebra System (CAS)
* Singular or sounder, a group of boar ...
cardinals, and has many more applications as well. The abbreviation "PCF" stands for "possible
cofinalities".
Main definitions
If ''A'' is an infinite set of
regular cardinal
In set theory, a regular cardinal is a cardinal number that is equal to its own cofinality. More explicitly, this means that \kappa is a regular cardinal if and only if every unbounded subset C \subseteq \kappa has cardinality \kappa. Infinit ...
s, ''D'' is an
ultrafilter on ''A'', then
we let
denote the cofinality of the ordered set of functions
where the ordering is defined as follows: