Definable set

Abstract: We discuss measures, invariant measures on definable groups, and genericity, often in an NIP (failure of the independence property) environment. We complete the proof of the third author’s conjectures relating definably compact groups G in saturated o-minimal structures to compact Lie groups. We also prove some other structural results about such G, for example the existence of a left invariant finitely additive probability measure on definable subsets of G. We finally introduce a new notion “compact domination” (domination of a definable set by a compact space) and raise some new conjectures in the o-minimal case.

Abstract: Publisher Summary This chapter describes the dynamics of model theory and some threats of set theory It is often risky to predict movements in logic It is noted that something remains of forcing and reduced products in categorical model theory, which provides universe and semantics more relevant than the set-theoretic ones A clear tendency is to focus on definability rather than decidability or structure theory Most of the decidability results identify definable sets, and relations, specifically equivalence relations A major contemporary theme is the topological structure of definable sets Kreisel had stressed the gigantic difference in importance between algebraic topology and set-theoretic topology The latter is ubiquitous in routine arguments and formulations, but the former is effective in advancing mathematical understanding Applied model theory is often concerned with number theory or analytic function theory

Abstract: In this paper we introduce the notion of a first order topological structure, and consider various possible conditions on the complexity of the definable sets in such a structure, drawing several consequences thereof. Our aim is to develop, for a restricted class of unstable theories, results analogous to those for stable theories. The “material basis” for such an endeavor is the analogy between the field of real numbers and the field of complex numbers, the former being a “nicely behaved” unstable structure and the latter the archetypal stable structure. In this sense we try here to situate our work on o-minimal structures [PS] in a general topological context. Note, however, that the p-adic numbers, and structures definable therein, will also fit into our analysis. In the remainder of this section we discuss several ways of studying topological structures model-theoretically. Eventually we fix on the notion of a structure in which the topology is “explicitly definable” in the sense of Flum and Ziegler [FZ]. In §2 we introduce the hypothesis that every definable set is a Boolean combination of definable open sets. In §3 we introduce a “dimension rank” on (closed) definable sets. In §4 we consider structures on which this rank is defined, and for which also every definable set has a finite number of definably connected definable components. We show that prime models over sets exist under such conditions.