Mathematical Sciences: Recursion Theory and Set Theory

三名教师调查员和他们的研究生参与了这个项目。马丁和斯蒂尔将在集合论中工作。在某种程度上，斯蒂尔和马丁将专注于内部模型理论和强确定性形式。马丁还希望完成他关于确定性证明的专著。Moschovakis将致力于算法理论及其在数学计算机科学基础中的应用。集合论是所有数学的基础。重要的问题涉及所有数字系统、代数、几何和分析背后的公理。某些公理是必要的吗？它们在特定的语境中是否不相关？在这方面特别重要的是所谓的确定性公理，它可以被框定为一个声明，即在特定的双人游戏中，一个玩家或另一个玩家具有获胜策略。这个博弈涉及到构建一个集合，让两个玩家轮流选择其中的成员。

Three faculty investigators and their graduate students are involved in this project. Martin and Steel will work in set theory. Steel, and Martin to some extent, will concentrate on inner model theory and on strong forms of determinacy. Martin also expects to complete his monograph on proofs of determinacy. Moschovakis will work in the theory of algorithms and its applications to the foundations of mathematical computer science. Set theory lies at the foundation of all mathematics. Important questions concern the axioms that underly all number systems, algebra, geometry and analysis. Are certain axioms necessary? Are they irrelevant in certain contexts? Particularly significant in this regard is the so-called Axiom of Determinacy, which can be framed as a statement that a certain two-person game has a winning strategy for one player or the other. The game in question involves constructing a set by having the two players alternate in selecting members for it.