Categoricity and u-rank in excellent classes

Journal of Symbolic Logic 68 (4):1317-1336 (2003)
Abstract
Let K be the class of atomic models of a countable first order theory. We prove that if K is excellent and categorical in some uncountable cardinal, then each model is prime and minimal over the basis of a definable pregeometry given by a quasiminimal set. This implies that K is categorical in all uncountable cardinals. We also introduce a U-rank to measure the complexity of complete types over models. We prove that the U-rank has the usual additivity properties, that quasiminimal types have U-rank 1, and that the U-rank of any type is finite in the uncountably categorical, excellent case. However, in contrast to the first order case, the supremum of the U-rank over all types may be ω (and is not achieved). We illustrate the theory with the example of free groups, and Zilber's pseudo analytic structures
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DOI 10.2178/jsl/1067620189
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References found in this work BETA
Finite Diagrams Stable in Power.Saharon Shelah - 1970 - Annals of Mathematical Logic 2 (1):69-118.
Strong Splitting in Stable Homogeneous Models.Tapani Hyttinen & Saharon Shelah - 2000 - Annals of Pure and Applied Logic 103 (1-3):201-228.
Ranks and Pregeometries in Finite Diagrams.Olivier Lessmann - 2000 - Annals of Pure and Applied Logic 106 (1-3):49-83.
Generalizing Morley's Theorem.Tapani Hyttinen - 1998 - Mathematical Logic Quarterly 44 (2):176-184.

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Citations of this work BETA
Simplicity and Uncountable Categoricity in Excellent Classes.Tapani Hyttinen & Olivier Lessmann - 2006 - Annals of Pure and Applied Logic 139 (1):110-137.
Forking in Short and Tame Abstract Elementary Classes.Boney Will & Grossberg Rami - 2017 - Annals of Pure and Applied Logic 168 (8):1517-1551.

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