PCF Theory - Unsolved Problems

Unsolved Problems

The most notorious conjecture in pcf theory states that |pcf(A)|=|A| holds for every set A of regular cardinals with |A|A). This would imply that if ℵ_ω 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|A), then pcf(A) has no inaccessible limit point. This is equivalent to the statement that pcf(pcf(A))=pcf(A).

Famous quotes related to unsolved problems:

“The child knows only that he engages in play because it is enjoyable. He isn’t aware of his need to play—a need which has its source in the pressure of unsolved problems. Nor does he know that his pleasure in playing comes from a deep sense of well-being that is the direct result of feeling in control of things, in contrast to the rest of his life, which is managed by his parents or other adults.”
—Bruno Bettelheim (20th century)

“Play permits the child to resolve in symbolic form unsolved problems of the past and to cope directly or symbolically with present concerns. It is also his most significant tool for preparing himself for the future and its tasks.”
—Bruno Bettelheim (20th century)