Mathematical Sciences: Forcing and O#

数学科学：强迫和 O

• 批准号: 9122320
• 负责人: Maurice Stanley
• 金额: $ 5万
• 依托单位: San Jose State University Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1992
• 资助国家: 美国
• 起止时间: 1992-07-15 至 1995-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9122320&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Forcing; O#

The investigator intends to continue work on generic objects constructed from 0 and other problems in forcing. In particular, he will attempt to extend S. D. Friedman's work, obtaining a non-constructible ZF-Pi-1-2-singleton constructible from 0. Several related problems in set theory will also be considered. 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. 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.

研究人员打算继续进行有关一般物体的工作 从0和其他问题的强制构造。 在 特别是，他将试图扩展S。D.弗里德曼的工作， 获得不可构造的ZF-Pi-1-2-单例可构造 从0。 集合论中的几个相关问题也将在 考虑了 集合论提供了一种流行的方法， 对于所有的数学来说，最著名的系统性尝试是 这就是罗素和怀特黑德的《数学原理》 重要的基本问题涉及独立性和 用于建立集合论的公理的一致性。 策梅罗 和Fraenkel的公理（简称ZF）是最重要的公理之一。 方便的基本公理集。 然而，据了解， 自从库尔特·哥德尔在20世纪30年代的工作以来， 集合论可以是完备的也可以是相容的。 这意味 没有一套公理能够强大到足以证明 可能的命题或其否定，但不能两者兼而有之。 后 这一惊人的定理已经建立了一个丰富的理论， 可能命题P，使得P或非P可以被加 而不会导致矛盾。 任何这样的提议 P被称为与ZF无关，并且可以被视为 集合论的附加公理 主要技术为 找到这样的独立命题是保罗·J·科恩的， 所谓的强迫方法及其后代。 这就是那个圆圈 参与和激励研究者研究的想法。