Special ultrafilters and cofinal subsets of $({}^\omega \omega , <^*)$

Abstract

The interplay between ultrafilters and unbounded subsets of ${}^\omega \omega$ with the order $<^*$ of strict eventual domination is studied. Among the tools are special kinds of non-principal (“free”) ultrafilters on $\omega$. These include simple P-points; that is, ultrafilters with a base that is well-ordered with respect to the reverse of the order $\subset ^*$ of almost inclusion. It is shown that the cofinality of such a base must be either $\mathfrak {b}$, the least cardinality of $<^*$-unbounded (“undominated”) set, or $\mathfrak {d}$, the least cardinality of a $<^*$-cofinal (“dominating”) set. The small uncountable cardinal $\pi \mathfrak {p}$ is introduced. Consequences of $\mathfrak {b}< \pi \mathfrak {p}$ and of $\mathfrak r < \mathfrak {d}$ are explored; in particular, both imply $\mathfrak {b}< \mathfrak {d}$. Here $\mathfrak r$ is the reaping number, and is also the least cardinality of a $\pi$-base for a free ultrafilter. Both of these inequalities are shown to occur if there exist simple P-points of different cofinalities (in other words, if $\mathfrak {b}< \mathfrak {d}$ and there exist simple $P_\mathfrak {b}$-points and $P_\mathfrak {d}$-points), but this is a long-standing open problem. Six axioms on nonprincipal ultrafilters on $\omega$ and the relationships between them are discussed along with various models of set theory in which one or more are known to hold (or are known to fail). The strongest of these, Axiom 1, is that for every free ultrafilter $\mathcal U$ and for every $<^*$-unbounded $<^*$-chain C of increasing functions in ${}^\omega \omega$, C is also unbounded in the ultraproduct ${}^\omega \omega , <_\mathcal U$. 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.