Thursday, March 15, 2018, from 4 to 5:30pm
East Hall, room 3088
Speaker: David J. Fernández Bretón (University of Michigan)
Title: Models of set theory with union ultrafilters and small covering of meagre, II
Union ultrafilters are ultrafilters that arise naturally from Hindman’s finite unions theorem, in much the same way that selective ultrafilters arise from Ramsey’s theorem, and they are very important objects from the perspective of algebra in the Cech–Stone compactification. The existence of union ultrafilters is known to be independent from the ZFC axioms (due to Hindman and Blass–Hindman), and is known to follow from a number of set-theoretic hypothesis, of which the weakest one is that the covering of meagre equals the continuum (this is due to Eisworth). In the first part of this two-talk series I exhibited a model of ZFC with union ultrafilters whose covering of meagre is strictly less than the continuum, obtained by means of a short countable support iteration. In this second talk, I will exhibit two more such models, one obtained by means of a countable support iteration of proper forcings, and the other by means of a single-step forcing (modulo being able to obtain an appropriate ground model).