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

斯考克罗夫特在数理逻辑方面的工作将集中于构造性实代数、p-adic 域的模型理论以及证明理论在代数中的应用。 斯考克罗夫特在实代数中找到了对有限逻辑复杂性的命题的构造性真实性的有效检验后，希望为这些构造性真实的命题分离出一组方便的公理。 给定这样的公理，他希望通过模型理论的方式证明构造性实代数中的定理。 斯考克罗夫特还想了解达纳·斯科特的直觉分析模型中的一阶实代数是否是递归不可判定的。 最近对亚分析集的研究可能使斯考克罗夫特能够将其对构造性真理的有效检验扩展到涉及某些先验函数的命题。 在p进场的模型理论中，斯考克罗夫特计划研究可定义的等价关系的性质、通常用于描述p进场的一阶语言之间表达能力的差异，以及Kochen积分定理有理函数定理的可能扩展。 斯考克罗夫特希望应用证明论论证来获得科亨定理给出的恒等式大小的原始递归或初等界限。

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.