Obviously, pcf(A) consists of regular cardinals. Considering ultrafilters concentrated on elements of A, we get that . Shelah proved, that if, then pcf(A) 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(A) such that . Consequently, . 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
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(A) 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(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 ℵω+1, which settles an old conjecture.
Read more about this topic: PCF Theory
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