2019/01/04 by Eskew, Monroe
#FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.1901.02822
Generalizing Keisler's notion of regularity for ultrafilters, Taylor introduced degrees of regularity for ideals and showed that a countably complete nonregular ideal on ω1 must be somewhere ω1-dense. We prove a dichotomy about degrees of regularity for κ-complete ideals on successor cardinals κ and apply this to show that Taylor's Theorem does not generalize to higher cardinals. In particular, the existence of a nonregular ideal on ω2 does not imply the existence of an ω2-dense ideal on ω2. We obtain similar results for normal ideals on \mathcal Pκ(λ).