The interplay between ultrafilters and unbounded subsets of \ with the order \ of strict eventual domination is studied. Among the tools are special kinds of non-principal ultrafilters on \. These include simple P-points; that is, ultrafilters with a base that is well-ordered with respect to the reverse of the order \ of almost inclusion. It is shown that the cofinality of such a base must be either \, the least cardinality of \-unbounded set, or \, the least cardinality of (...) a \-cofinal set. The small uncountable cardinal \ is introduced. Consequences of \ and of \ are explored; in particular, both imply \. Here \ is the reaping number, and is also the least cardinality of a \-base for a free ultrafilter. Both of these inequalities are shown to occur if there exist simple P-points of different cofinalities and there exist simple \-points and \-points), but this is a long-standing open problem. Six axioms on nonprincipal ultrafilters on \ and the relationships between them are discussed along with various models of set theory in which one or more are known to hold. The strongest of these, Axiom 1, is that for every free ultrafilter \ and for every \-unbounded \-chain C of increasing functions in \, C is also unbounded in the ultraproduct \. The other axioms replace one or both quantifiers with “there exists.” The negation of Axiom 3 in a model provides a family of normal sequentially compact spaces whose product is not countably compact. The question of whether such a family exists in ZFC, even with “normal” weakened to “regular”, is a famous unsolved problem of set-theoretic topology, known as the Scarborough–Stone problem. (shrink)

We prove that a number of axioms, each a consequence of PFA (the Proper Forcing Axiom) are equivalent. In particular we show that TOP (the Thinning-out Principle as introduced by Baumgartner in the Handbook of set-theoretic topology), is equivalent to the following statement: If I is an ideal on ω 1 with ω 1 generators, then there exists an uncountable $X \subseteq \omega_1$ , such that either [ X] ω ∩ I = ⊘ or $\lbrack X\rbrack^\omega \subseteq I$.

Reviewed Works:Andreas Blass, Saharon Shelah, Ultrafilters with Small Generating Sets.Andreas Blass, Saharon Shelah, There May Be Simple $P_{\aleph_1}$- and $P_{\aleph_2}$-Points and the Rudin-Keisler ordering may be downward directed.Andreas Blass, Near Coherence of Filters. II: Applications to Operator Ideals, the Stone- Cech Remainder of a Half-Line, Order Ideals of Sequences, and the Slenderness of Groups.Andreas Blass, Saharon Shelah, Near Coherence of Filters III: A Simplified Consistency Proof.Andreas Blass, Claude Laflamme, Consistency Results About Filters and the Number of Inequivalent Growth Types.Andreas Blass, (...) J. Steprans, S. Watson, Applications of Superperfect Forcing and Its Relatives.Andreas Blass, Saharon Shelah, Ultrafilters with Small Generating Sets. (shrink)