PCF theory


PCF theory is the name of a mathematical theory, introduced by Saharon, 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".

Main definitions

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
where the ordering is defined as follows.
if.
pcf is the set of cofinalities that occur if we consider all ultrafilters on A, that is,

Main results

Obviously, pcf consists of regular cardinals. Considering ultrafilters concentrated on elements of A, we get that
. Shelah proved, that if, then pcf has a largest element, and there are subsets of A such that for each ultrafilter D on A, is the least element θ of pcf such that. Consequently,.
Shelah also proved that if A is an interval of regular cardinals, then pcf is also an interval of regular cardinals and |pcf|<|A|+4.
This implies the famous inequality

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 holds for every ultrafilter D with B∈D. Then J<λ is the ideal generated by the sets. There exist scales, i.e., for every λ∈pcf there is a sequence of length λ of elements of which is both increasing and cofinal mod J<λ. This implies that the cofinality of under pointwise dominance is max.
Another consequence is that if λ is singular and no regular cardinal less than λ is Jónsson, then also λ+ is not Jónsson. In particular, there is a Jónsson algebra on ℵω+1, which settles an old conjecture.

Unsolved problems

The most notorious conjecture in pcf theory states that |pcf|=|A| holds for every set A of regular cardinals with |A|ω is strong limit, then the sharp bound

holds. The analogous bound

follows from Chang's conjecture or even from the nonexistence of a Kurepa tree.
A weaker, still unsolved conjecture states that if |A|limit point. This is equivalent to the statement that pcf=pcf.

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.