Tameness in Expansions of the Real Field

What might it mean for a first-order expansion of the field of real numbers to be tame or well behaved? In recent years, much attention has been paid by model theorists and real-analytic geometers to the o-minimal setting: expansions of the real field in which every definable set has finitely many connected components. But there are expansions of the real field that define sets with infinitely many connected components, yet are tame in some well-defined sense (e.g., the topological closure of every definable set has finitely many connected components, or every definable set has countably many connected components). The analysis of such structures often requires a mixture of model-theoretic, analytic-geometric and descriptive set-theoretic techniques. An underlying idea is that first-order definability, in combination with the field structure, can be used as a tool for determining how complicated is a given set of real numbers. Throughout, m and n range over N (the non-negative integers). Given a first-order structure M with underlying set M , “definable” (in M) means “definable in M with parameters from M” unless otherwise noted. If no ambient space M is specified, then “definable set” means “definable subset of some M”. I use “reduct” and “expansion” in the sense of definability, that is, given structures M1 and M2 with common underlying set M , I say that M1 is a reduct of M2—equivalently, M2 is an expansion of M1, or M2 expands M1—if every set definable in M1 is definable in M2. For the most part, we shall be concerned with the definable sets of a structure, so we identify M1 and M2 if they are interdefinable (that is, each is a reduct of the other). An expansion M of a dense linear order (M,<) is o-minimal if every definable subset of M is a finite union of points and open intervals. From now on, R denotes a fixed, but arbitrary, expansion of the real line (R, <). “Definable” means “definable in R” unless noted otherwise. The real field (R,+, · ) is denoted by R. The sequel consists of two parts: Part 1 is mostly expository and somewhat informal; technical details and proofs are mostly deferred to Part 2. General references for background include: van den Dries [7] (a model-theoretic survey of o-minimality) and [10] (a text on topological o-minimality, with essentially no model theory); van den Dries and Miller [11] (focussing on the analytic geometry of o-minimal expansions of R); and anything along the lines of Hausdorff [17], Kechris [18], Kuratowski [19] and Oxtoby [27]. Please note: I attempt neither to cite only original sources nor to provide an historical survey. 2000 Mathematics Subject Classification. Primary 03C64; Secondary 28A05, 54H05. To appear in Logic Colloquium ’01. *Please discard any earlier drafts of this paper. A nontrivial change has been made to the July 14, 2002 version: The definition of “special submanifold” has been weakened.