Ultrafilters on a countable set

An ultrafilter on a set is a proper collection of subsets of ' that set which is maximal among such collections having the finite intersection property. Ultrafilters were popularized by N.Bourbaki for their use in describing topological convergence, but for some time there was little discussion of the possible structural properties that an indiwdual ultrafilter might possess. This paper is concerned only with ultrafilters on a countable s,~t and a method of describing them by building them from certain minimal ultrafilters; most of the work here has co~l-e from Chapters 1, 2, and 4 of [ I ]. The first part of this paper describes a cer• tain tree of ultrafilters and the third part describes an ordering in which this tree is embedded; the remaining two parts deat with tninimal ultrafilters and with products of uitrafilters. Theorems and lemmas are always thought of +her~, as being proved in Zermelo-Fraenkel set theory with the axiom of choice, ZFC. If a theorem T is to be proved in some other set theory, say the theory 17, it is stated in ZFC in the following manner: