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Completeness And Categoricity (In Power): Formalization Without Foundationalism.

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journal contribution
posted on 22.01.2016, 00:00 by J.T. Baldwin
Abstract. We propose a criterion to regard a property of a theory (in first or second order logic) as virtuous: the property must have significant mathematical consequences for the theory (or its models). We then rehearse results of Ajtai. Marek. Magidor. H. Friedman and Solovay to argue that for second order logic, 'categoricity' has little virtue. For first order logic, categoricity is trivial: but 'categoricity in power' has enormous structural consequences for any of the theories satisfying it. The stability hierarchy extends this virtue to other com plete theories. The interaction of model theory and traditional mathematics is examined by considering the views of such as Bourbaki. Hrushovski. Kazhdan. and Shelah to flesh out the argument that the main impact of formal methods on mathematics is using formal definability to obtain results in 'mainstream' mathematics. Moreover, these methods (e.g. the stability hierarchy) provide an organization for much mathematics which gives specific content to dreams of Bourbaki about the architecture of mathematics.

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Publisher Statement

This is a copy of an article published in the Bulletin of Symbolic Logic © 2014 Association for Symbolic Logic and Cambridge University Press Publications.

Publisher

Association for Symbolic Logic (Cambridge University Press)

issn

1079-8986

Issue date

01/03/2014

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