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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 \(T\).

MSC:

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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References:

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