On definable subsets of p-adic fields

Author:

MacIntyre Angus

Abstract

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 π as part of our basic data. The problem is that in the presence of π 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.

Publisher

Cambridge University Press (CUP)

Subject

Logic,Philosophy

Cited by 40 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. ON GROUPS WITH DEFINABLE F-GENERICS DEFINABLE IN P-ADICALLY CLOSED FIELDS;The Journal of Symbolic Logic;2023-06-09

2. An undecidability result for the asymptotic theory of p-adic fields;Annals of Pure and Applied Logic;2023-02

3. Éz fields;Journal of Algebra;2023-01

4. Definably topological dynamics of p-adic algebraic groups;Annals of Pure and Applied Logic;2022-04

5. Model theory of adeles I;Annals of Pure and Applied Logic;2022-03

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