Wim Veldman And

Wc establish constructive refinements of several well-known theorems in elementary model theory. The additive group of the real numbers may be embedded elementarily into the additive group of pairs of real numbers, constructively as well as classically. Introduction. Intuitionistic model theory, as we understand it, is part of intuitionistic mathematics. We study intuitionistic structures from the model-theoretic point of view in an intuitionistic way. We are not trying to find non-intuitionistic interpretations of formally intuitionistic theories. The paper is divided into eight sections. In Section 1 we notice that notions such as “elementary equivalence” and “elementary substructure” have a straightforward constructive meaning. We classify formulas according to their quantifier-depth and define corresponding refinements of the basic model-theoretic concepts. We introduce strongly homogeneous structures, that is, structures with the property that every local isomorphism extends to an automorphism of the structure. We also introduce the weaker notion of a back-and-forth-homogeneous structure. We prove a theorem that will help us to find elementary substructures of back-and-forthhomogeneous structures. In Section 2 we recapitulate the intuitionistic construction of the continuum and prove that the structure (R, <) is strongly homogeneous. In Section 3, we consider subsets A of M such that (A} <) is an elementary substructure of (R, <). We recover and extend the most important results of [7]. In Section 4, we prove that intuitionistic Baire space (^ ,# o ) (the universal spread), considered as a set with an apartness relation, not with an order relation, is strongly homogeneous, and we mention some applications of this result. Section 5 is our first intuitionistic intermezzo. We discuss some consequences of the continuity principle, and show that the apartness structure (R, #) is not strongly homogeneous, In Section 6 we prove that (R ,#) is back-and-forth-homogeneous. In Section 7, our second intuitionistic intermezzo, we show that Fraisse’s characterization of elementary equivalence is not valid constructively. In Section 8 we study (R, +), the additive group of the real numbers, and consider several structures that are elementarily equivalent to (R, +). Most of our proofs, although intuitionistically correct, may count as “classical” proofs of the corresponding “classical” theorems. Nobody will find fault with our avoidance of indirect arguments. In the proof of Theorem 3.3.4 we use a version of the axiom of countable choice. This version of the axiom of countable choice is Received March 15, 1994; revised November 21, 1994, and July 5,1995. © 1996, Association for Symbolic Logic 0022-4812/96/6103-0002/S3.30