Mathematical Sciences: Set Theory

数学科学：集合论

• 批准号: 9205530
• 负责人: Sy Friedman
• 金额: --
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing grant
• 财政年份: 1992
• 资助国家: 美国
• 起止时间: 1992-08-15 至 1996-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9205530&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Set; Theory

Friedman intends to investigate four areas within set theory in addition to beginning work on a book on the method of L-coding. (The latter should help make the many significant applications of this method accessible to a wider audience of set-theorists.) (1) With his former student, Dorshka Wylie (now an NSF fellow and an instructor at M.I.T.), he is engaged in a radically new approach to the core model K for a strong cardinal, which should provide a powerful new tool for establishing combinatorial principles in K. (2) A second goal is to enable coding methods to lift smoothly to K, permitting extension of results such as his proof of the Pi-1-2- Singleton Conjecture to the core model context. (3) A long- standing open question in the theory of iterated forcing concerns the problem of extending the method of countable support iteration beyond omega-2 without collapsing cardinals. A number of important consistency problems could be solved by such a technique. With his student, Dimitiros Tzimas, Friedman will attempt to develop such a method through the use of morasses. (4) Friedman will also explore questions in descriptive set theory as to whether the full structure theory for projective sets can be developed directly from the existence of large cardinals, without the use of infinite game theory. Set theory provides the popular way to lay the foundations for all of mathematics, the best known systematic attempt to do this being Russell and Whitehead's Principia Mathematica, dating from the early part of the 20th Century. Important foundational questions concern the independence and the consistency of the axioms used to establish set theory. Zermelo and Fraenkel's axioms (ZF for short) are one of the most convenient sets of basic axioms. However, it has been known since the work of Kurt Goedel in the 1930's that no axioms for set theory can be complete as well as consistent. This means that no set of axioms can be powerful enough to prove every possible proposition or else its negation, but not both. Upon this startling theorem has been erected a rich theory, treating possible propositions P such that either P or Not P can be added to ZF without resulting in a contradiction. Any such proposition P is said to be independent of ZF and can be taken as an additional axiom of set theory. The principal technique for finding such independent propositions is Paul J. Cohen's so-called method of forcing and its offspring. This is the circle of ideas involved in and motivating the investigator's research.

除了开始写一本关于 L 编码方法的书之外，弗里德曼还打算研究集合论中的四个领域。 （后者应该有助于使这种方法的许多重要应用为更广泛的集合理论家所接受。）（1）与他以前的学生 Dorshka Wylie（现在是 NSF 研究员和麻省理工学院的讲师）一起，他正在研究一种针对强基数的核心模型 K 的全新方法，这应该为在 K 中建立组合原理提供一个强大的新工具。(2) 第二个目标是启用编码方法 顺利提升到 K，允许将结果（例如 Pi-1-2-单例猜想的证明）扩展到核心模型上下文。 (3) 迭代强迫理论中一个长期存在的悬而未决的问题涉及将可数支持迭代方法扩展到 omega-2 之外而不折叠基数的问题。 许多重要的一致性问题可以通过这种技术来解决。 弗里德曼将与他的学生迪米蒂罗斯·齐马斯一起尝试通过使用沼泽来开发这种方法。 (4)弗里德曼还将探讨描述性集合论中的问题，即射影集的完整结构理论是否可以直接从大基数的存在中发展而来，而不需要使用无限博弈论。 集合论提供了为所有数学奠定基础的流行方法，最著名的系统性尝试是罗素和怀特海的数学原理，其历史可以追溯到 20 世纪初。 重要的基础问题涉及用于建立集合论的公理的独立性和一致性。 策梅洛和弗兰克尔公理（简称ZF）是最方便的基本公理集之一。 然而，自从库尔特·哥德尔 (Kurt Goedel) 在 1930 年代的工作以来，人们就知道集合论的公理不可能是完整且一致的。 这意味着没有一组公理能够强大到足以证明每个可能的命题或其否定，但不能同时证明两者。 在这个令人震惊的定理的基础上，建立了一个丰富的理论，处理可能的命题 P，使得 P 或 Not P 可以添加到 ZF 而不会导致矛盾。 任何这样的命题 P 都被认为是独立于 ZF 的，并且可以被视为集合论的附加公理。 寻找这种独立命题的主要技术是保罗·J·科恩所谓的强制方法及其衍生方法。 这是参与并激励研究者研究的思想圈。