Editing PCF theory (section)

== Main results ==
Obviously, pcf(''A'') consists of regular cardinals. Considering ultrafilters concentrated on elements of ''A'', we get that
<math>A\subseteq \operatorname{pcf}(A)</math>. Shelah proved, that if <math>|A|<\min(A)</math>, then pcf(''A'') has a largest element, and there are subsets <math>\{B_\theta:\theta\in \operatorname{pcf}(A)\}</math> of ''A'' such that for each ultrafilter ''D'' on ''A'', <math>\operatorname{cf}\left(\prod A/D\right)</math> is the least element θ of pcf(''A'') such that <math>B_\theta\in D</math>. Consequently, <math>\left|\operatorname{pcf}(A)\right|\leq2^{|A|}</math>.
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''|<sup>+4</sup>.
This implies the famous inequality
<div style="text-align: center;"><math>2^{\aleph_\omega}<\aleph_{\omega_4}</math></div>
assuming that ℵ<sub>ω</sub> is [[limit cardinal|strong limit]].

If λ is an infinite cardinal, then ''J''<sub><λ</sub> is the following ideal on ''A''. ''B''∈''J''<sub><λ</sub> if <math>\operatorname{cf}\left(\prod A/D\right)<\lambda</math> holds for every ultrafilter ''D'' with ''B''∈''D''. Then ''J''<sub><λ</sub> is the ideal generated by the sets <math>\{B_\theta:\theta\in \operatorname{pcf}(A),\theta<\lambda\}</math>. There exist ''scales'', i.e., for every λ∈pcf(''A'') there is a sequence of length λ of elements of <math>\prod B_\lambda</math> which is both increasing and cofinal mod ''J''<sub><λ</sub>. This implies that the cofinality of <math>\prod A</math> under pointwise dominance is max(pcf(''A'')).
Another consequence is that if λ is singular and no regular cardinal less than λ is [[Jónsson cardinal|Jónsson]], then also λ<sup>+</sup> is not Jónsson. In particular, there is a [[Jónsson algebra]] on ℵ<sub>ω+1</sub>, which settles an old conjecture.