Completeness and Categoricity (in power): Formalization without Foundationalism

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 complete 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. 1 What is the role of categoricity? In correspondence in 2008, Michael Detlefsen raised a number of questions about the role of categoricity. We discuss two of them in this paper. Question I1: (A) Which view is the more plausible—that theories are the better the more nearly they are categorical, or that theories are the better the more they give rise to significant non-isomorphic interpretations? ∗I realized while writing that the title was a subconscious homage to the splendid historical work on Completeness and Categoricity by Awodey and Reck [6]. 1The first were questions III.A and III.B in the original letter. The second was question IV in the Detlefsen letter. I thank Professor Detlefsen for permission to quote this correspondence.