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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Totally categorical structures


Author: Ehud Hrushovski
Journal: Trans. Amer. Math. Soc. 313 (1989), 131-159
MSC: Primary 03C45; Secondary 03C35
DOI: https://doi.org/10.1090/S0002-9947-1989-0943605-1
MathSciNet review: 943605
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Abstract: A first order theory is totally categorical if it has exactly one model in each infinite power. We prove here that every such theory admits a finite language, and is finitely axiomatizable in that language, modulo axioms stating that the structure is infinite. This was conjectured by Vaught. We also show that every $ {\aleph _0}$-stable, $ {\aleph _0}$-categorical structure is a reduct of one that has finitely many models in small uncountable powers. In the case of structures of disintegrated type we nearly find an explicit structure theorem, and show that the remaining obstacle resides in certain nilpotent automorphism groups.


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DOI: https://doi.org/10.1090/S0002-9947-1989-0943605-1
Keywords: Categoricity, stability, finite axiomatizability, coordinatization by projective space, strongly minimal sets
Article copyright: © Copyright 1989 American Mathematical Society