Mathemtical Sciences: RUI: Problems in Forcing

数学科学：RUI：强迫问题

• 批准号: 9505157
• 负责人: Maurice Stanley
• 金额: $ 7.68万
• 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: 1995
• 资助国家: 美国
• 起止时间: 1995-09-15 至 1998-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9505157&HistoricalAwards=false
• 关键词: Mathemtical; Sciences; RUI; Problems; Forcing

9505157 Stanley The project involves continued work on class forcing and non-generic reals, some combinatorial questions regarding stationary sets that have application to unique generic objects, and other problems in forcing. 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 set 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 consistent as well as complete. 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 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 axioms 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. ***

小行星9505157 该项目涉及继续工作的类强迫和非通用reals，一些组合的问题，关于固定套有应用程序的独特的通用对象，以及其他问题的强迫。 集合论为所有数学奠定了基础，最著名的系统性尝试是罗素和怀特海的《数学原理》。重要的基础问题涉及的独立性和一致性的公理用于建立集理论。 Zermelo和Fraenkel公理（简称ZF）是最方便的基本公理之一。然而，自从库尔特·哥德尔在20世纪30年代的工作以来，人们已经知道，集合论的公理不可能是一致的和完整的。这意味着没有一组公理可以强大到足以证明每一个可能的命题或其否定，但不能两者兼而有之。在这个令人吃惊的定理上，已经建立了一个丰富的理论，处理可能的命题P，使得P或非P可以被添加到ZF而不会导致矛盾。任何这样的命题都被认为独立于ZF，并且可以被视为集合论的附加公理。寻找这种独立公理的主要技术是保罗·科恩的所谓的强迫方法及其后代。这是参与和激励研究者研究的思想循环。 ***