JUSSI KETONEN OPEN PROBLEMS IN THE THEORY OF UL TRAFILTERS INTRODUCTION 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 foHows that of the most recent set-theoretic literature; for example Ixl denotes the cardinality of the set x, small Greek letters lX, [3, 'Y, .•. denote ordinals, cardinals are initial ordinals, the set Yx or xY 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: I. 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: zED, Y :2 z -+ Y E D Xl ... Xn E D -+ Xl " ••• "Xn E D ZED-+z# 0 (b) D is maximal: Z s: x -+ ZED or x - zED D is non-principal if in addition (c) YEX-+{Y} fj=D. It is sometimes useful to think of an ultrafilter D as a finitely additive measure p attaining only values 0 and I: Define p(z) = 1 if zED and 0 otherwise. The notion of an ultrafilter gained some attention from the Polish mathematicians of the 30's, and it eventually became an object of study in general topology: 227 J. Hintikka. I. Niiniluoto, and E. Saarinen (eds.), Essays on Mathematical and Philosophical Logic, 227-247. All Rights Reserved. Copyright © 1978 by D. Reidel Publi,hing Company, Dordrecht, Holland.