Mathematical Sciences: Logical Issues in Constructive Mathematics, and Metamathematics of Algebra

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Scowcroft's work, in mathematical logic, will center on constructive real algebra, the model theory of p-adic fields, and applications of proof theory to algebra. Having found an effective test for the constructive truth of propositions, in real algebra, of a bounded logical complexity, Scowcroft hopes to isolate a convenient set of axioms for these constructively true propositions. Given such axioms, he would like to prove theorems in constructive real algebra by model- theoretic means. Scowcroft also wants to learn whether first- order real algebra in Dana Scott's model for intuitionistic analysis is recursively undecidable. Recent work on subanalytic sets may allow Scowcroft to extend his effective test for constructive truth to propositions involving certain transcendental functions. In the model theory of p-adic fields, Scowcroft plans to study the properties of definable equivalence relations, the differences in expressive power between the first-order languages commonly used to describe a p-adic field, and possible extensions of Kochen's theorem on integral-definite rational functions. Scowcroft hopes to apply proof-theoretic arguments to obtain primitive-recursive or elementary bounds for the size of the identities given by Kochen's theorem.