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
denote the cofinality of the ordered set of functions
pcf(A) is the set of cofinalities that occur if we consider all ultrafilters on A, that is, Obviously, pcf(A) consists of regular cardinals.
, then pcf(A) has a largest element, and there are subsets
This implies the famous inequality assuming that ℵω is strong limit.
Then J<λ is the ideal generated by the sets
There exist scales, i.e., for every λ∈pcf(A) there is a sequence of length λ of elements of
which is both increasing and cofinal mod J<λ.
under pointwise dominance is max(pcf(A)).
In particular, there is a Jónsson algebra on ℵω+1, which settles an old conjecture.
The most notorious conjecture in pcf theory states that |pcf(A)|=|A| holds for every set A of regular cardinals with |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| 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.