Mathematical Sciences: Problems in Large Cardinals, Forcing and Combinatorics

数学科学：大基数、强迫和组合问题

• 批准号: 8703433
• 负责人: Richard Laver
• 金额: $ 8.18万
• 依托单位: University of Colorado at Boulder
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1987
• 资助国家: 美国
• 起止时间: 1987-07-01 至 1991-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8703433&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Problems; Large; Cardinals

The area of research is set theory, in particular, large cardinals and forcing. Set theory may be viewed as containing mathematics, that is, every theorem of classical mathematics may be formulated as a statement about sets, and then formally derived from the standard collection ZFC of axioms for set theory. Beginning with the work of Godel (1936) and Cohen (1963), set theorists have shown that certain problems, on which mathematicians have worked, are independent, i.e. neither provable nor disprovable from ZFC. The first and most well known example of this is the Godel-Cohen result that Cantor's continuum hypothesis is independent. Some statements, independent of ZFC, are nevertheless known to be provable when ZFC is augmented by axioms asserting the existence of large infinite cardinal numbers. This project has two parts: firstly, to continue the study of a class of very large cardinals, and secondly, to study the consistency and independence of von Neumann's conjecture and a collection of related problems. Such studies deepen our understanding of the limits of mathematical proof.

研究领域是集合论，特别是大 红衣主教和强迫。 集合论可以被看作包含 数学，也就是说，古典数学的每一个定理都可以 被公式化为关于集合的陈述，然后正式地 从集合公理的标准集合ZFC导出 理论 从哥德尔（1936年）和科恩的工作开始， （1963），集合理论家已经表明，某些问题， 数学家们都工作过，都是独立的，也就是说， 从ZFC可以证明也可以证明。 第一个也是最著名的 这方面的例子是哥德尔-科恩的结果，康托的连续统， 假设是独立的 一些独立于ZFC的声明， 然而，当ZFC被扩充为 断言存在大无穷基数的公理 号码 该项目有两个部分：第一，继续 研究一类非常大的红衣主教，其次，研究 冯·诺依曼猜想的一致性和独立性， 相关问题的集合。 这些研究加深了我们 理解数学证明的局限性。