Application of infinitary languages to metric spaces
Ralph D. Kopperman · Pacific Journal of Mathematics · 1967
We attempt to lay the groundwork for applying the recently-developed theory of models for the infinitary languages Ll ε to analysis.It will be shown that within one of these languages, axioms may be written whose class of models is precisely the metric spaces.We show that two complete separable metric spaces are elementarily equivalent in this language if and only if they are isomorphic and obtain an elimination of quantifiers for such spaces.A method is developed for transferring results on metric spaces to structures with metrics whose relations are closed under the metric topology.This class includes Banach Spaces.When not otherwise indicated, definitions, notations, and modeltheoretic results used in this paper may be found in [5],The paper will be divided into three sections, as follows: I. Axiomatization of metric spaces, II.Theorems on metric systems, and III.Metric algebraic systems.Model-theoretic results will be introduced as needed.I* Axiomatizatίon of metric spaces* Our method of axiomatization of metric spaces is chosen in order to make their definition possible with one sentence in L™ lCϋ (for the definition of L κβ , ω lf and ω, see [5]).We begin with a definition of the rational numbers: DEFINITION 1.1.Q = , a structure of type q - will be called a rational number system if and only if:(1) Q is an ordered field (see [9, pp.77, 5]).(2) (Vv o )(O<v o -+V i<ω \/ j<ω (Vo a+ ... +1) = 1+ ... +1)).j factors i factors THEOREM 1.2.Each rational number system is isomorphic to the rational numbers.