The Real Line in Elementary Submodels of Set Theory

It is a truism that all of mathematics can be expressed in the language of set theory, i.e. the predicate calculus (including equality) with the single two-place relation ∈. As with any other countable language, the language of set theory is subject to the Löwenheim-Skolem Theorem, which asserts that if we have some axioms – e.g. the usual axioms of set theory (ZFC) – formulated in this language and those axioms have any model at all, then they have models of all infinite sizes. Those models will have varying portions of the real line included in them; the question we are interested in here is what can we say about these subsets of R. In particular, whether any two such subsets of the same size are isomorphic as topological spaces, linear orders, or fields. We first need to do some preliminary work to make these questions precise. First of all there is the standard difficulty that, by Gödel’s incompleteness theorem, we cannot be assured that there is any model of the axioms of set theory, and certainly not one which is a set rather than a class. The standard solution is to argue that any mathematical proof only involves finitely many axioms of set theory and one can prove within set theory the existence of nice models for these, namely H(θ), θ an uncountable cardinal, H(θ) the collection of sets which have cardinality less than θ, whose members have cardinality less than θ, whose members of members have cardinality less than θ, etc. Any particular proof in practice only uses sets up to a certain level in the set-theoretic hierarchy, e.g. reals, sets of reals, and sets of sets of reals, so one can work with H(θ), θ “sufficiently large”, rather than the entire set-theoretic universe V . The non-logician reader will not lose any insight, and will indeed gain some, by replacing every occurrence of H(θ) in the rest of the paper by V . Now given H(θ), satisfying some fragment of the axioms of set theory, by the LöwenheimSkolem Theorem, given any non-empty subset S of H(θ) we can close off, i.e. throw in witnesses for existential quantifiers, so as to produce an elementary submodel M of H(θ) of size |S| + א0, i.e. a subset M of H(θ), S ⊆ M , such that M as a collection of sets with the membership relation that 1 Supported by NSF 2 Supported by NSERC grant A7354. 1991 Mathematics Subject Classification. Primary 03C62, 03E55, 06A05, 03C55, 03C55, 54F05; Secondary 12L99.