## 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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