Notes for the Tenth Appalachian Set Theory Workshop From the lectures given by Andreas Blass Summary: Notes for the Tenth Appalachian Set Theory Workshop From the lectures given by Andreas Blass On the topics of ultrafilters and cardinal characteristics of the continuum Delivered on Jan. 30, 2010 at the Miami University in Oxford, Ohio Compiled by Nicholas Rupprecht 1. Ultrafilters Definition 1. An ultrafilter on the set X is a U P(X) such that (1) U, (2) X U, (3) A B and A U implies B U, (4) A,B U implies A B U, (5) For all A X, A U or X - A U, (6) A B U implies A U or B U. Remark 2. Some of these are redundant. The first four alone define a filter on X. For (4) and (6), the converses are also true by (3). (5) may also be read as A U X - A U. We can think of this as: membership in U respects boolean combinations (propositional connec- tives). In this way, we can view U as a map from 2X to 2 such that for any operation on 2k (and the operation it induces on (2X)k), U Collections: Mathematics