Mathematical Sciences: Recursion Theory and Set Theory

9505153 Martin Martin将研究集合论和描述性集合论，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从大基本公理证明确定性的研究专著。Steel还将在集合理论和描述性集合理论中进行一般的工作，更具体地说，是关于满足指定的大基数或确定性假设的ZFC的规范内部模型的构建和研究。Moschovakis将致力于不动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，以及非确定性和并发设置中递归方程的规范解的构建。在过去的25年里，该基金的主要研究者一直是重塑经典数学集合论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，这些假设使得解决许多关于实数的长期开放问题成为可能。马丁和斯蒂尔将继续研究这一基础项目，寻求对其进行扩展，并划定其局限性。莫斯科瓦基斯大约在十年前改变了兴趣，他目前的研究项目旨在为计算机科学提供一个严谨的数学基础。他的工作的一个主要的、最终的目标是构建一个“程序的逻辑”，它将在计算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. ***