JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. Association for Symbolic Logic is collaborating with JSTOR to digitize, preserve and extend access to The Journal of Symbolic Logic. ?0. The brilliant work of Ax-Kochen [1], [2], [3] and Ersov [6], and later work by Kochen [7] have made clear very striking resemblances between real closed fields and p-adically closed fields, from the model-theoretical point of view. Cohen [5], from a standpoint less model-theoretic, also contributed much to this analogy.In this paper we shall point out a feature of all the above treatments which obscures one important resemblance between real and p-adic fields. We shall outline a new treatment of the p-adic case (not far removed from the classical treatments cited above), and establish an new analogy between real closed and p-adically closed fields.We want to describe the definable subsets of p-adically closed fields.
Tarski [9] in his pioneering work described the first-order definable subsets of real closed fields. Namely, if K is a real-closed field and X is a subset of K first-order definable on K using parameters from K then X is a finite union of nonoverlapping intervals (open, closed, half-open, empty or all of K). In particular, if X is infinite, X has nonempty interior. Now, there is an analogous question for p-adically closed fields. If K is p-adically closed, what are the definable subsets of K? To the best of our knowledge, this question has not been answered until now.What is the difference between the two cases? Tarski's analysis rests on elimination of quantifiers for real closed fields. Elimination of quantifiers for p-adically closed fields has been achieved [3], but only when we take a cross-section iT as part of our basic data. The problem is that in the presence of iT it becomes very difficult to figure out what sort of set is definable by a quantifier free formula. We shall see later that use of the cross-section increases the class of definable sets.Our motivation is largely topological. Both real closed and p-adically closed fields are topological fields, and we are mainly interested in the topological character of the definable sets. In particular, does an infinite definable set have nonempty interior? The answer is yes, if we do not allow iT in our definitions. The answer is no, if we do allow iT.It is known that in the set-up of [3] we cannot eliminate quantifiers in the pure language of valued fields. There seems to be no example in the literature. We give one in Appendix 1.
