Open Problems in the Theory of Ultrafilters

Abstract

The purpose of this paper is to present a list of open questions in the theory of ultrafilters. Most of them seem almost impenetrable by the usual methods of set-theory. Needless to say, the list of such questions is infinite, and the topics chosen for this paper reflect the personal tastes and prejudices of the author. Our notation and terminology follows that of the most recent set-theoretic literature; for example |x|denotes the cardinality of the set x, small Greek letters α,β,γ,… denote ordinals, cardinals are initial ordinals, the set y x or x y denotes the set of all functions y → x etc. For more, we refer the reader to Mathias [20]. By an ultrafilter over a set x we mean here a maximal filter in the field of subsets of x; that is: 1. Definition: D is an ultrafilter over a set x if D is a collection of subsets of x so that (a) D is a filter: $$z \in D,y \supseteq z \to y \in D$$ $${x_1} \ldots {x_n} \in D \to {x_1} \cap \ldots \cap {x_n} \in $$ $$z \in D \to z \ne 0$$ (b) D is maximal: $$z \subseteq x \to z \in Dorx - z \in D$$ D is non-principal if in addition (c) $$y \in x \to \left\{ y \right\} \notin D.$$