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 $\operatorname\left(\prod A/D\right)$ denote the cofinality of the ordered set of functions
$\prod A$ where the ordering is defined as follows:
f if $\\in D$.
pcf(''A'') is the set of cofinalities that occur if we consider all ultrafilters on ''A'', that is,

$\operatorname(A)=\left\.$

Main results

Obviously, pcf(''A'') consists of regular cardinals. Considering ultrafilters concentrated on elements of ''A'', we get that
$A\subseteq \operatorname(A)$. Shelah proved, that if $, A, <\min(A)$, then pcf(''A'') has a largest element, and there are subsets $\$ of ''A'' such that for each ultrafilter ''D'' on ''A'', $\operatorname\left(\prod A/D\right)$ is the least element θ of pcf(''A'') such that $B_\theta\in D$. Consequently, $\left, \operatorname(A)\\leq2^$.
Shelah also proved that if ''A'' is an interval of regular cardinals (i.e., ''A'' is the set of all regular cardinals between two cardinals), then pcf(''A'') is also an interval of regular cardinals and , pcf(''A''), <, ''A'', ⁺⁴.
This implies the famous inequality

$2^<\aleph_$

assuming that ℵ_ω is strong limit.

If λ is an infinite cardinal, then ''J''_<λ is the following ideal on ''A''. ''B''∈''J''_<λ if $\operatorname\left(\prod A/D\right)<\lambda$ 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(''A'') there is a sequence of length λ of elements of $\prod B_\lambda$ which is both increasing and cofinal mod ''J''_<λ. This implies that the cofinality of $\prod A$ under pointwise dominance is max(pcf(''A'')).
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 ℵ_ω+1, which settles an old conjecture.

Unsolved problems

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

$2^<\aleph_$

holds. The analogous bound

$2^<\aleph_$

follows from Chang's conjecture ( Magidor) or even from the nonexistence of a Kurepa tree

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