A countable dense homogeneous set of reals of size א

We prove there is a countable dense homogeneous subspace of R of size א1. The proof involves an absoluteness argument using an extension of the Lω1ω(Q) logic obtained by adding predicates for Borel sets. A separable topological space X is countable dense homogeneous (CDH) if, given any two countable dense subsets D and D′ of X, there is a homeomorphism h of X such that h[D] = D′. The main purpose of this note is to show the following. Theorem 1. There is a countable dense homogeneous set of reals X of size א1. Moreover , X can be chosen to be a λ-set. Recall that a set of reals is a λ-set if all of its countable subsets are relatively Gδ. Theorem 1 and the fact that a λ-set cannot be completely metrizable solve problems 390 and 389 from [5]. A construction of a CDH metric space that is not completely metrizable necessarily uses some form of the Axiom of Choice. In [8] it was shown that under sufficient large cardinal assumptions every CDH metric space in L(R) is completely metrizable. Our proof of Theorem 1 uses Keisler’s completeness theorem for the Lω1ω(Q) logic (see §2), and a secondary purpose of this note is in explicitly stating a somewhat general method for proving absoluteness of the existence of an uncountable set of reals whose properties are described using Borel sets as parameters. 2000 Mathematics Subject Classification: 54E52, 54H05, 03E15.