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在构造实代数、p进域的模型论以及证明论在代数中的应用。在发现了对有界逻辑复杂性的实代数命题的构造真值的有效检验后，Scowcroft希望为这些构造真值命题分离出一组方便的公理。有了这样的公理，他想用模型论的方法证明构造实代数中的定理。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递归地不可判定。最近关于子分析集的工作可能允许Scowcroft将构造真值的有效检验扩展到涉及某些超越函数的命题。在p进域的模型论中，Scowcroft计划研究可定义等价关系的性质，常用的描述p进域的一阶语言之间表达能力的差异，以及Kochen定理在整定有理函数上的可能推广。Scowcroft希望应用证明论的论证来得到由Kochen定理给出的恒等式的大小的原初-递归或初等边界。

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.