Good Ideals in Fields of Sets

Abstract

In this paper we shall study some problems which are purely settheoretical in character, but which arose in connection with a problem in the theory of models. The principal results of this paper form an important part of the proofs of the series of results in the theory of models which are announced in [1], [2], [3], and in the appendix of [4]. The present article, however, is entirely free of any notions from the theory of models. It is well-known that, for every infinite cardinal a, there exist nonprincipal prime ideals in the field S(X) of all subsets of a set X of power a, and furthermore there are exactly 221 such prime ideals. The object of this paper is to show that non-principal prime ideals with certain additional properties exist. Let X be a set of power a, and let S( Y) be the set of all non-empty finite subsets of a set Y. If f, g map S.( Y) into the field of subsets of X, we write g > f if, for all a E S.( Y), g(a) -f(a). Let us say that an ideal I in the field S(X) is SJ( Y)-good if, for every monotonic function f on S,(Y) into I, there exists an additive function g > f on S,( Y) into I. If Y is countable, then every ideal I is S.( Y)-good (cf. Corollary 4.2). Our two main results are as follows.