A complicated \(\omega \)-stable depth 2 theory. (English) Zbl 1215.03052

This paper investigates the connection between the model-theoretic notion of eni-depth and the descriptive set-theoretic notion of Borel reducibility. Eni-depth is a technical model-theoretic notion used in describing the complexity of countable models of a theory. Given a countable language \(T\) and an \(\mathcal{L}\)-theory \(T\), the space of countable models of \(T\), denoted \(X_T\), is a standard Borel space and the isomorphism relation between models of \(T\), denoted \(\cong_T\), is an analytic equivalence relation on \(X_T\). Given two analytic equivalence relations \(E\) and \(F\) on standard Borel spaces \(X\) and \(Y\), we say that \(E\) Borel reduces to \(F\), denoted \(E\leq_B F\), if there is a Borel map \(f:X\to Y\) such that, for all \(a,b\in X\), \((a,b)\in E \Leftrightarrow (f(a),f(b))\in F\); intuitively, \(E\leq_B F\) means that \(E\) is no more complicated than \(F\). It thus makes sense to consider \(\leq_B\) with regard to equivalence relations of the form \(\cong_T\) in order to gauge the difficulty of the classification of models of \(T\) up to isomorphism.
In the author’s thesis, he showed that theories \(T\) with eni-depth \(1\) are smooth, that is \(\cong_T\) Borel-reduces to the equality relation on \(r\). Since smooth relations are uncomplicated, eni-depth 1 guarantees that \(\cong_T\) is not very complicated. The current paper asks what are the implications having eni-depth equal to 2. The answer is that eni-depth 2 no longer guarantees that \(\cong_T\) is smooth. In fact, the author constructs a model-theoretically well-behaved theory \(T\) (e.g., \(T\) is complete, admits QE, is \(\omega\)-stable, has NDOP) with eni-depth \(2\) such that \(\cong_T\) is not a Borel relation. Since smooth equivalence relations are always Borel, this shows that \(\cong_T\) is not smooth. (The author raises the question of whether \(\cong_T\) for the \(T\) under discussion is Borel-complete, which means that it is as complicated as possible.) The author shows that \(\cong_T\) is not Borel by proving an equivalent property, namely that there is no countable ordinal bound on the Scott heights of all countable models of \(T\).


03C45 Classification theory, stability, and related concepts in model theory
03C15 Model theory of denumerable and separable structures
03E15 Descriptive set theory
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