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性数学和p-进领域的模型理论上。斯考克罗夫特发现了实代数中逻辑复杂性有限的命题构造性真实性的有效检验，他希望为这些构造性真命题分离出一组方便的公理。他还想知道达纳·斯科特的直观分析模型中的一阶实代数是否递归不可判定。斯考克罗夫特直观分析的新模型可能允许他发展一阶实代数的新模型，而最近关于次分析集的工作可能允许他将他对构造性真理的有效检验扩展到涉及某些超越函数的命题。在p-adad场的模型理论中，斯考克罗夫特计划研究可定义等价关系的性质，通常用于描述p-adic场的一阶语言在表达能力上的差异，以及Macintyre的p-adics的一阶理论的量词消除过程的效率。代数基础中的这些问题使现代数理逻辑涉及到另一个相关领域。这类综合一直是我们这个时代一些最重要的数学的来源。

Scowcroft's work, in mathematical logic, will center on constructive mathematics and the model-theory of p-adic fields. 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. He also wants to learn whether first-order real algebra in Dana Scott's model for intuitionistic analysis is recursively undecidable. Scowcroft's new model for intuitionistic analysis may permit him to develop a new model for first-order real algebra, and recent work on subanalytic sets may allow him 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 the efficiency of quantifier-elimination procedures for Macintyre's first-order theory of the p-adics. These issues in the foundations of algebra bring modern mathematical logic to bear upon another related area. Syntheses of this sort have been the source of some of the most significant mathematics of our time.