Abstract
We solve the classification problem and essentially the spectrum problem for universal theories (see [6] for discussion of the meaning of this). We first solve it forT such that ifM 1,M 2 elementarily extendM 0 and are independent over it, then overM 0 ∪M 1 there is a prime model. This generalizes [2]. This was subsequently used and generalized for countable first order theories. (This will appear in [5].) But note that there the theory is countable and in the case of structure the model is prime over a non-forking tree of models; here the model is generated by the union (and theT not necessarily countable). The universality is used in
Theorey.If T is stable and complete then either (A)for every M 1<M (l=0, 1, 2)models of T, if M 0 ⊆M 1,M 2, {M 1,M 2}is independent over M 0 (i.e. tp(M 1,M 2)is finitely satisfiable in M 0),then the submodel of M which M 1 ⋃M 2 generates is an elementary submodel of M, or (B)there is an unstable theory extending the universal part of T (we can replace universal by Σ2 and slightly more).
Conclusion. For any universalT:Either (a) for every modelM ofT there is a treeI with ≦ω levels and submodelsN η (η ∈I) of power ≦2|T| (by [5], just ≦|T|) such that (i)M is generated by ∪ηεl N η, (ii)η <v⇒⇒N η, (iii) ifv is an immediate successor ofη then tp(N v, ⋃{{N p:ρ ∈I,v≨ϱ}) is finitely satisfiable inN η (note that asking this just for quantifier-free formulas is enough).Or (b) for every cardinalλ>|T|, there are 2γ non-isomorphic models for powerλ.
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References
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Dedicated to Professor Abraham Robinson
The author would like to thank John Baldwin for the interesting talks in September 1980 which led to §3 of this work, Rami Grossberg for various corrections, and the BSF and NSF for their partial support.
This paper was originally intended to appear in the Proceedings of the Model Theory Year at the Institute for Advanced Studies, The Hebrew University of Jerusalem, September 1980 — August 1981, published in Isr. J. Math., Vol. 49, Nos. 1–3, 1984.
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Shelah, S. The spectrum problem III: Universal theories. Israel J. Math. 55, 229–256 (1986). https://doi.org/10.1007/BF02801997
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DOI: https://doi.org/10.1007/BF02801997