Measurability Properties on Small Cardinals
- Author(s): Eskew, Monroe Blake
- Advisor(s): Zeman, Martin
- et al.
Ulam proved that there cannot exist a probability measure on the reals for which every set is measurable and gets either measure zero or one. He asked how large a collection of partial 0-1 valued measures is required so that every set of reals is measurable in one of them. Alaoglu and Erdos proved that if the continuum hypothesis holds, then countably many measures is not enough, and Ulam asked if aleph_1 many can suffice. This question was shown to be independent of ZFC by Prikry and Woodin. Here, we examine the analogous questions on successor cardinals above aleph_1 and on spaces of the form P_kappa(lambda). We generalize Woodin's consistency results to these contexts, producing models of ideals of minimal density on various spaces starting from models of almost-huge cardinals. We show some interactions between these ideals, cardinal arithmetic, and square principles. Then we show that certain characterizations of a positive answer to Ulam's question, namely the existence of dense ideals and nonregular ideals, are equivalent on aleph_1 but not for higher cardinals. Some tension appears in separating these properties while preserving the GCH, but we show this is possible using structures we call "coherent forests," about which we show several results of independent interest. The main result is that if almost-huge cardinals are consistent, then ZFC+GCH does not prove that the existence of dense and nonregular ideals is equivalent for successor cardinals above aleph_1. Our methods also lead to a new result on the individual consistency but collective inconsistency of some types of generic large cardinals.