On almost precipitous ideals (Q964452)

A regular uncountable cardinal \(\kappa\) is \textit{almost precipitous} if for each ordinal \(\tau < {(2^\kappa)}^+\), there is a \(\kappa\)-complete ideal \(I\) over \(\kappa\) with the property that every generic ultrapower of \(I\) is well-founded up to the image of \(\tau\). The paper presents a wealth of results concerning almost precipitousness and related notions (semi-precipitousness, precipitousness and pseudo-precipitousness). Here are three of these results: {\parindent=7mm \begin{itemize}\item[(A)] ``There exists an almost precipitous cardinal'' and ``There exists an almost precipitous cardinal with normal ideals witnessing its almost precipitousness'' are equiconsistent; \item[(B)] Assume \(0^\#\), and let \(\eta\) be a regular cardinal in \(L\). Then there is a generic extension \(L[G]\) of \(L\) such that (a) \(L[G]\) and \(L\) have the same cardinals less than or equal to \(\eta\), and (b) in \(L[G]\), \(\eta^+\) is almost precipitous as witnessed by normal filters; \item[(C)] If there are class many Ramsey cardinals, then any uncountable cardinal is almost precipitous as witnessed by normal filters. \end{itemize}}