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".
Contents
Main definitions
If A is an infinite set of regular cardinals, D is an ultrafilter on A, then we let
Main results
Obviously, pcf(A) consists of regular cardinals. Considering ultrafilters concentrated on elements of A, we get that
assuming that ℵω is strong limit.
If λ is an infinite cardinal, then J<λ is the following ideal on A. B∈J<λ if
Unsolved problems
The most notorious conjecture in pcf theory states that |pcf(A)|=|A| holds for every set A of regular cardinals with |A|<min(A). This would imply that if ℵω is strong limit, then the sharp bound
holds. The analogous bound
follows from Chang's conjecture (Magidor) or even from the nonexistence of a Kurepa tree (Shelah).
A weaker, still unsolved conjecture states that if |A|<min(A), then pcf(A) has no inaccessible limit point. This is equivalent to the statement that pcf(pcf(A))=pcf(A).
Applications
The theory has found a great deal of applications, besides cardinal arithmetic. The original survey by Shelah, Cardinal arithmetic for skeptics, includes the following topics: almost free abelian groups, partition problems, failure of preservation of chain conditions in Boolean algebras under products, existence of Jónsson algebras, existence of entangled linear orders, equivalently narrow Boolean algebras, and the existence of nonisomorphic models equivalent in certain infinitary logics.
In the meantime, many further applications have been found in Set Theory, Model Theory, Algebra and Topology.