Notre Dame Journal of Formal Logic 50 (4):365-380 (2009)

Abstract
In "A proof of Vaught's conjecture for ω-stable theories," the notions of ENI-NDOP and eni-depth have been introduced, which are variants of the notions of NDOP and depth known from Shelah's classification theory. First, we show that for an ω-stable first-order complete theory, ENI-NDOP allows tree decompositions of countable models. Then we discuss the relationship between eni-depth and the complexity of the isomorphism relation for countable models of such a theory in terms of Borel reducibility as introduced by Friedman and Stanley and construct, in particular, a sequence of complete first-order ω-stable theories $(T_\alpha)_{\alpha < \omega_1}$ with increasing and cofinal eni-depth and isomorphism relations which are strictly increasing with respect to Borel reducibility
Keywords omega-stability   classifications   countable models   Borel reducibility
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DOI 10.1215/00294527-2009-016
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