Event Detail

We intend to use specific combinatorial problems concerning ultrafilters to illustrate the title. Indecomposable ultrafilters were introduced by Keisler and Prikry as a weakening of measures on measurable cardinals, loosely speaking, by not insisting on countable completeness. Silver asked whether an inaccessible cardinal carrying an indecomposable ultrafilter necessarily has to be measurable. Sheard answered this negatively. However, recently Goldberg showed Silver’s question has a positive answer above a strongly compact cardinal. We will show that strong forcing axioms, for example PFA, imply a positive answer to Silver’s question, giving evidence to the heuristic that strong forcing axioms assert ω2 is “supercompact”. Previous evidence includes failure of square principles and the singular cardinal hypothesis. Then we will define a family of weak indexed square principles and use them to demonstrate that the positive result we obtain is indeed optimal, thus demonstrating an extent of the heuristic and showing that ω2 is “mortal” after all.

Joint work with Chris Lambie-Hanson and Assaf Rinot.