Categoricity and u-rank in excellent classes

Journal of Symbolic Logic 68 (4):1317-1336 (2003)
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
Keywords No keywords specified (fix it)
Categories (categorize this paper)
DOI 10.2178/jsl/1067620189
 Save to my reading list
Follow the author(s)
My bibliography
Export citation
Find it on Scholar
Edit this record
Mark as duplicate
Revision history
Request removal from index
Download options
Our Archive

Upload a copy of this paper     Check publisher's policy     Papers currently archived: 28,756
Through your library
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.

Add more references

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.

Add more citations

Similar books and articles
The Geometry of Forking and Groups of Finite Morley Rank.Anand Pillay - 1995 - Journal of Symbolic Logic 60 (4):1251-1259.
The Morley Rank of a Banach Space.José Iovino - 1996 - Journal of Symbolic Logic 61 (3):928-941.
The Mitchell Order Below Rank-To-Rank.Itay Neeman - 2004 - Journal of Symbolic Logic 69 (4):1143 - 1162.
Fields of Finite Morley Rank.Frank Wagner - 2001 - Journal of Symbolic Logic 66 (2):703-706.
On the Number of Models of Uncountable Theories.Ambar Chowdhury & Anand Pillay - 1994 - Journal of Symbolic Logic 59 (4):1285-1300.
A Model and its Subset.Ludomir Newelski - 1992 - Journal of Symbolic Logic 57 (2):644-658.
Supersimple Ω-Categorical Groups and Theories.David M. Evans & Frank O. Wagner - 2000 - Journal of Symbolic Logic 65 (2):767-776.
Rank and Dimension in Difference-Differential Fields.Ronald F. Bustamante Medina - 2011 - Notre Dame Journal of Formal Logic 52 (4):403-414.
Geometry of *-Finite Types.Ludomir Newelski - 1999 - Journal of Symbolic Logic 64 (4):1375-1395.

Monthly downloads

Added to index


Total downloads

17 ( #286,975 of 2,178,142 )

Recent downloads (6 months)

1 ( #316,663 of 2,178,142 )

How can I increase my downloads?

My notes
Sign in to use this feature

There  are no threads in this forum
Nothing in this forum yet.

Other forums