Neoclosed forcing

Summary

Abstract. A general model-theoretic theory of approximation is presented which encompasses approximation methods found in analysis in both standard and nonstandard settings. We first give a simple version of the main idea, in the classical metric space setting. This was inspired by work of Anderson and Henson. We inductively define the notions of a closed formula, closed forcing, and the set of approximations of a closed formula. It is shown that given a relatively compact sequence, a closed formula is forced if and only if all its approximations are eventually true, and also if and only if the formula is true at every limit point. Then, in the nonstandard setting, we prove harder analogous results using our theory of neometric spaces, where saturation arguments take the place of compactness arguments. These results shed light on well-known nonstandard constructions that produce new theorems about standard objects.

Introduction. One of the main uses of model theory outside of mathematical logic itself has been the introduction in the early sixties of nonstandard analysis by Abraham Robinson (see [18]). He showed how to apply nonstandard models of the appropriate language to a wide variety of problems in analysis. His construction captured the attention of mathematicians because it made the old idea of infinitesimal quantities available to modern mathematics (for a detailed history of the development of these ideas see the last chapter in [18]).

Robinson's original presentation, which relied heavily on the theory of types, has been “cleaned up”, so that today one does not have to be a logician in order to understand and use nonstandard analysis. Nevertheless there are close ties between model theory and developments that have originated from nonstandard practice. The purpose of this paper is to develop one of these ties: we give a general model theoretic theory of approximation which encompasses approximation methods found in both standard and nonstandard settings.