A complicated ω-stable depth 2 theory

Abstract

We present a countable complete first order theory T which is model theoretically very well behaved: it eliminates quantifiers, is ω-stable, it has NDOP and is shallow of depth two. On the other hand, there is no countable bound on the Scott heights of its countable models, which implies that the isomorphism relation for countable models is not Borel.

Borel complexity and potential canonical Scott sentences

“…K is the Koerwien theory, originating in [14] and defined in Section 6. Koerwien proved that K is complete, ω-stable, eni-NDOP, and of eni-depth 2.…”

“…Using these tools, we analyze the complexity of the class of countable models of four complete, first-order theories T for which (Mod(T ), ∼ =) is properly analytic, yet admit very different behavior. We prove that both 'Binary splitting, refining equivalence relations' and Koerwien's example [14] of an eni-depth 2, ω-stable theory have (Mod(T ), ∼ =) non-Borel, yet neither is Borel complete. We give a slight modification of Koerwien's example that also is ω-stable, eni-depth 2, but is Borel complete.…”

A descriptive Main Gap Theorem

“…Indeed, ∼ = ω T is Borel if and only if there is an "effective" procedure which, using only countable set-theoretical operations such as unions, intersections, and complements, allows us to determine whether two countable models of T are isomorphic or not -in other words, there is a Borel procedure to classify the countable models of T up to isomorphism. Unfortunately, there is no relation between Shelah's classification of T in terms of its stability properties and the simplicity of ∼ = ω T in the descriptive set-theoretic sense: for example, the theory of dense linear orders is unstable, but the isomorphism relation on its countable models is very simple (it is a Borel equivalence relation with Borel rank 2 and only 4 different classes); conversely, in [Koe11] it is shown that there are theories T which are very simple stability-wise, but such that ∼ = ω T is not even Borel. This failure forces us to move to the uncountable setting again.…”

Borel $$^{*}$$ Sets in the Generalized Baire Space and Infinitary Languages

“…is Borel(κ) (in particular, ISO(T, κ) is Borel(κ) for the theory T constructed by Koerwien in [Koe11] and ISO(DLO, κ) is not Borel(κ), in fact not even ∆ 1 1 (κ), see Section 1). For more on such questions, see [FHK11] and [HK12].…”

“…Then in Shelah's classification DLO is a very complicated theory but since DLO is ω-categorical ISO(DLO, ω) is very simple, Borel and of very low rank. On the other hand, M. Koerwien has shown in [Koe11] that there is an ω-stable NDOP theory T of depth 2 such that ISO(T, ω) is not Borel. In Shelah's classification ω-stable NDOP theories of depth 2 are considered very simple.…”