Mathematical Sciences: Recursion Theory and Set Theory

9505153马丁·马丁将从事集合论和描述性集合论的工作，特别是确定性假设，他的第一个目标是完成他期待已久的关于大型基数公理的确定性证明的研究专著。斯蒂尔还将在集合论和描述性集合论中发挥作用，更具体地说，是构建和研究满足特定大型基数或确定性假设的规范的ZFC内部模型。Moschovakis将致力于定点递归及其在计算理论中的应用，包括递归算法的数学建模、递归定义的逻辑以及在不确定和并发环境下递归方程的规范解的构造。在过去的二十五年里，该基金的首席研究人员一直是重塑经典的集合论数学基础的关键贡献者之一，他们引入和研究了关于集合的新假设，使解决许多关于实数的长期悬而未决的问题成为可能。马丁和钢铁公司将继续致力于这一基本计划，寻求扩大它，并勾勒出它的局限性。Moschovakis在大约十年前改变了兴趣，他目前的研究计划旨在为计算机科学提供严谨的数学基础。他工作的主要和最终目标之一是构建一种“程序逻辑”，这将在计算机系统的开发和验证中有用。***

9505153 Martin Martin will work in set theory and descriptive set theory, especially determinacy hypotheses, and his first goal is to complete his long-awaited research monograph on proofs of determinacy from large cardinal axioms. Steel will also work generally in set theory and descriptive set theory, and more specifically on the construction and study of canonical, inner models of ZFC that satisfy specified large cardinal or determinacy assumptions. Moschovakis will work on fixpoint recursion and its applications to the theory of computation, including the mathematical modeling of recursive algorithms, the logic of recursive definitions, and the construction of canonical solutions for recursive equations in non-deterministic and concurrent settings. The Principal Investigators of this Grant have been among the key contributors to the reshaping of the classical, set-theoretic foundations of mathematics during the last twenty-five years, with the introduction and study of new hypotheses about sets that have made it possible to solve many long-standing, open problems about the real numbers. Martin and Steel will continue working on this fundamental program, seeking to extend it and to delineate its limitations. Moschovakis changed interests about ten years ago, and his current research program aims to provide a rigorous, mathematical foundation for computer science. One of the main, ultimate aims of his work is to construct a "logic of programs" that will be useful in the development and verification of computing systems. ***