Seminar on Reckoning

There may be no I-ultrafilter for any F_sigma ideal I

The notion of I-ultrafilter was introduced by Baumgartner, and it has been useful in providing a framework for classifying many combinatorial properties of ultrafilters. In this talk we prove, however, that the class of F_sigma ideals may provide a vacuous classification of the combinatorial properties of ultrafilters, that is, F_sigma ideals may be not useful in providing combinatorial information about ultrafilters. This in turn implies that consistently there is no Hausdorff ultrafilter, thus answering a classical question of M. Di Nasso and M. Forti.