Borel Equivalence Relations and Lascar Strong Types

Abstract

The "space" of Lascar strong types, on some sort and relative to a given complete theory T , is in general not a compact Hausdorff topological space. We have at least three (modest) aims in this paper. The first is to show that spaces of Lascar strong types, as well as other related spaces and objects such as the Lascar group Gal L (T ) of T , have well-defined Borel cardinalities (in the sense of the theory of complexity of Borel equivalence relations). The second is to compute the Borel cardinalities of the known examples as well as of some new examples that we give. The third is to explore notions of definable map, embedding, and isomorphism, between these and related quotient objects. We also make some conjectures, the main one being roughly "smooth iff trivial" The possibility of a descriptive set-theoretic account of the complexity of spaces of Lascar strong types was touched on in the paper [2], where the first example of a "non G-compact theory" was given. The motivation for writing this paper is partly the discovery of new examples via definable groups, in [3], [4] and the generalizations in [6].

The Borel cardinality of Lascar strong types

“…In [9], this theorem is verified for many examples. We will actually prove a slightly stronger result (see Theorem 4.13).…”

“…The authors of [9] proved Borel cardinality goes, this does not depend on the model M , even when restricting to a KP strong type. (…”

“…It is therefore unclear to what category this quotient belongs. In [9], the authors suggest viewing it through the framework of descriptive set theory (this idea had already been mentioned in [3]). They formally interpret the notion of equality of Lascar strong types as a Borel equivalence relation over a compact Polish space, and then consider the position of this relation in the Borel reducibility hierarchy.…”

An embedding theorem of 𝔼0 with model theoretic applications

“…In this setting we will translate our relation E into an F σ relation on X M , as was done in [6]. For a countable model A ⊆ M, S α (M ) is Polish and if Y is as in Remark 3.8 then Y M is a Polish space (every G δ set is), and similarly to [3, Proposition 1.41] (with the same proof as there) we have the following proposition.…”

“…It turns out that if T and α are countable, one can interpret E as an (honest) F σ equivalence relation on a compact Polish space in a very natural way, which equips E with a well-defined Borel cardinality. This was done for Lascar strong types in [6], where many examples are computed, but in fact works for any E. It is explained in details in Secs. 3…”