We study the $\kappa $ -Borel-reducibility of isomorphism relations of complete first-order theories by using coloured trees. Under some cardinality assumptions, we show the following: For all theories T and T’, if T is classifiable and T’ is unsuperstable, then the isomorphism of models of T’ is strictly above the isomorphism of models of T with respect to $\kappa $ -Borel-reducibility.

Answering one of the main questions of [S.-D. Friedman, T. Hyttinen and V. Kulikov, Generalized descriptive set theory and classification theory, Mem. Amer. Math. Soc. 230 80, Chap. 7], we show that there is a tight connection between the depth of a classifiable shallow theory [Formula: see text] and the Borel rank of the isomorphism relation [Formula: see text] on its models of size [Formula: see text], for [Formula: see text] any cardinal satisfying [Formula: see text]. This is achieved by (...) establishing a link between said rank and the [Formula: see text]-Scott height of the [Formula: see text]-sized models of [Formula: see text], and yields to the following descriptive set-theoretical analog of Shelah’s Main Gap Theorem: Given a countable complete first-order theory [Formula: see text], either [Formula: see text] is Borel with a countable Borel rank, or it is not Borel at all. The dividing line between the two situations is the same as in Shelah’s theorem, namely that of classifiable shallow theories. We also provide a Borel reducibility version of the above theorem, discuss some limitations to the possible complexities of [Formula: see text], and provide a characterization of categoricity of [Formula: see text] in terms of the descriptive set-theoretical complexity of [Formula: see text]. (shrink)

We provide here the first steps toward a Classification Theory ofElementary Classes with no maximal models, plus some mild set theoretical assumptions, when the class is categorical in some λ greater than its Löwenheim-Skolem number. We study the degree to which amalgamation may be recovered, the behaviour of non μ-splitting types. Most importantly, the existence of saturated models in a strong enough sense is proved, as a first step toward a complete solution to the o Conjecture for these classes. Further (...) results are in preparation. (shrink)

We discuss dividing lines (in model theory) and some test questions, mainly the universality spectrum. So there is much on conjectures, problems and old results, mainly of the author and also on some recent results.

Let be an abstract elementary class with amalgamation, and Lowenheim Skolem number LS. We prove that for a suitable Hanf number gc0 if χ0 < λ0 λ1, and is categorical inλ1+ then it is categorical in λ0.

If T has only countably many complete types, yet has a type of infinite multiplicity then there is a c.c.c. forcing notion Q such that, in any Q-generic extension of the universe, there are non-isomorphic models M 1 and M 2 of T that can be forced isomorphic by a c.c.c. forcing. We give examples showing that the hypothesis on the number of complete types is necessary and what happens if `c.c.c.' is replaced by other cardinal-preserving adjectives. We also give (...) an example showing that membership in a pseudo-elementary class can be altered by very simple cardinal-preserving forcings. (shrink)

If T is an unstable theory of cardinality <λ or countable stable theory with OTOP or countable superstable theory with DOP, λω λω1 in the superstable with DOP case) is regular and λ<λ=λ, then we construct for T strongly equivalent nonisomorphic models of cardinality λ. This can be viewed as a strong nonstructure theorem for such theories. We also consider the case when T is unsuperstable and develop further a result of Shelah about the existence of L∞,λ-equivalent nonisomorphic models for (...) such T. In addition, we show that a natural analogue of Scott's isomorphism theorem fails for models of power κ, if κω is regular, assuming κ<κ=κ. (shrink)

We show in the paper that for any non-classifiable countable theory T there are non-isomorphic models and that can be forced to be isomorphic without adding subsets of small cardinality. By making suitable cardinal arithmetic assumptions we can often preserve stationary sets as well. We also study non-structure theorems relative to the Ehrenfeucht-Fraïssé game.

We prove the existence of pairs of models of the same cardinality which are very equivalent according to EF games, but not isomorphic. We continue the paper [4], but we do not rely on it. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim).

The results herein form part of a larger project to characterize the classification properties of the class of submodels of a homogeneous stable diagram in terms of the solvability (in the sense of [1]) of the potential isomorphism problem for this class of submodels. We restrict ourselves to locally saturated submodels of the monster model m of some power π. We assume that in Gödel's constructible universe ������, π is a regular cardinal at least the successor of the first cardinal (...) in which ������ is stable. We show that the collection of pairs of submodels in ������ as above which are potentially isomorphic with respect to certain cardinal-preserving extensions of ������ is equiconstructible with 0 # . As 0 # is highly "transcendental" over ������, this provides a very strong statement to the effect that potential isomorphism for this class of models not only fails to be set-theoretically absolute, but is of high (indeed of the highest possible) complexity. The proof uses a novel method that does away with the need for a linear order on the skeleton. (shrink)

We characterize the classifiability of a countable first-order theory T in terms of the solvability of the potential-isomorphism problem for models of T.