On the consistency strength of critical leaps

Abstract

In the analysis of the blurry \(\textsf{HOD}\) hierarchy, one of the fundamental concepts is that of a leap, and it turned out that critical leaps are of particular interest. A critical leap is a leap which is the cardinal successor of a singular strong limit cardinal. Such a leap is sudden if its cardinal predecessor is not a leap, and otherwise, it is smooth. In prior work, I showed that the existence of a sudden critical leap is equiconsistent with the existence of a measurable cardinal. Here, I show that if the cofinality of the cardinal predecessor of a sudden critical leap is required to be uncountable, the consistency strength increases considerably. I also show that when focusing on critical leaps whose cardinal predecessors have uncountable cofinality, the consistency strength of a smooth critical leap is much lower than that of a sudden critical leap. Finally, I observe that in contrast to the countable cofinality setting, \(\aleph _{\omega _1+1}\), e.g., cannot be a sudden critical leap.