The real line in elementary submodels of set theory

Abstract

The use of elementary submodels has become a standard tool in set-theoretic topology and infinitary combinatorics. Thus, in studying some combinatorial objects, one embeds them in a set, M, which is an elementary submodel of the universe, V (that is, (M; Є) ≺ (V; Є)). Applying the downward Löwenheim-Skolem Theorem, one can bound the cardinality of M. This tool enables one to capture various complicated closure arguments within the simple “≺”.However, in this paper, as in the paper [JT], we study the tool for its own sake. [JT] discussed various general properties of topological spaces in elementary submodels. In this paper, we specialize this consideration to the space of real numbers, ℝ. Our models M are not in general transitive. We will always have ℝ Є M, but not usually ℝ ⊆ M. We plan to study properties of the ℝ ⋂ M's. In particular, as M varies, we wish to study whether any two of these ℝ ⋂ M's are isomorphic as topological spaces, linear orders, or fields.As usual, it takes some sleight-of-hand to formalize these notions within the standard axioms of set theory (ZFC), since within ZFC, one cannot actually define the notion (M;Є) ≺ (V;Є). Instead, one proves theorems about M such that (M;Є) ≺ (H(θ);Є), where θ is a “large enough” cardinal; here, H(θ) is the collection of all sets whose transitive closure has size less than θ.