Summary
We prove here:
Theorem. LetT be a countable complete superstable non ω-stable theory with fewer than continuum many countable models. Then there is a definable groupG with locally modular regular generics, such thatG is not connected-by-finite and any type inG eq orthogonal to the generics has Morley rank.
Corollary. LetT be a countable complete superstable theory in which no infinite group is definable. ThenT has either at most countably many, or exactly continuum many countable models, up to isomorphism.
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Supported by NSF grant DMS 90-06628
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Low, L.F., Pillay, A. Superstable theories with few countable models. Arch Math Logic 31, 457–465 (1992). https://doi.org/10.1007/BF01277487
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DOI: https://doi.org/10.1007/BF01277487