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Notes on Quasiminimality and Excellence

Published online by Cambridge University Press:  15 January 2014

John T. Baldwin
John T. Baldwin*
Affiliation:
Department of Mathematics, Statistics and Computer Science, University of Illinoisat Chicago, USAE-mail: [email protected]
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Abstract

This paper ties together much of the model theory of the last 50 years. Shelah's attempts to generalize the Morley theorem beyond first order logic led to the notion of excellence, which is a key to the structure theory of uncountable models. The notion of Abstract Elementary Class arose naturally in attempting to prove the categoricity theorem for Lω1,ω(Q). More recently, Zilber has attempted to identify canonical mathematical structures as those whose theory (in an appropriate logic) is categorical in all powers. Zilber's trichotomy conjecture for first order categorical structures was refuted by Hrushovski, by the introducion of a special kind of Abstract Elementary Class. Zilber uses a powerful and essentailly infinitary variant on these techniques to investigate complex exponentiation. This not only demonstrates the relevance of Shelah's model theoretic investigations to mainstream mathematics but produces new results and conjectures in algebraic geometry.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 10 , Issue 3 , September 2004 , pp. 334 - 366
Copyright
Copyright © Association for Symbolic Logic 2004

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