Combinatorial Set Theory

组合集合论

• 批准号: 0757507
• 负责人: Justin Moore
• 金额: $ 44.49万
• 依托单位: Cornell University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-06-01 至 2013-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0757507&HistoricalAwards=false
• 关键词: Combinatorial; Set; Theory

Moore's work in infinite combinatorics concerns classification and basisproblems for uncountable structures and their connection with cardinalexponentiation. On one hand, strong axioms such as the Proper ForcingAxiom and Woodin's Pmax axiom are invoked to build embeddings andmorphisms between structures such as linear orders and topologicalspaces. Since these strong axioms themselves imply that the cardinalityof the real line is the second uncountable cardinal, it is natural toask whether the classification theorems which follow from them alreadyfix the cardinality of the continuum. The research supported by thisgrant aims to prove new classification results from these strong axioms,to better understand the relationship between these classificationresults and the value of the continuum, and to improve our understandingof the strong axioms themselves.The oldest results in set theory concern the rigorous development of the"size" of an infinite set. The cardinal numbers provide a linear scalewith which one can measure the number of elements a set has --- alsoknown as its cardinality. One of the earliest questions in set theorywas Cantor's Continuum Problem: Determine the cardinality of the realnumber line. In the 1960s, this problem was shown to be independent ofthe usual axioms of mathematics. Still, it is unclear whether someother compelling mathematical statement, also undecidable, might settlethe Continuum Problem. Moore's research aims to both establish theindependence of new classification results for infinite mathematicalstructures and to relate these results to the Continuum Problem.

摩尔的工作在无限组合关注的分类和基础问题的不可数结构和他们的联系与cardinalexponentiation。 一方面，强公理，如适当的强迫公理和Woodin的Pmax公理被用来建立嵌入和结构之间的态射，如线性序和拓扑空间。 由于这些强公理本身意味着真实的线的基数是第二个不可数基数，因此很自然地要问，从它们得出的分类定理是否已经确定了连续统的基数。 该基金支持的研究旨在证明这些强公理的新分类结果，以更好地理解这些分类结果与连续统值之间的关系，并提高我们对强公理本身的理解。集合论中最古老的结果涉及无限集合“大小”的严格发展。 基数提供了一个线性尺度，人们可以用它来度量一个集合中元素的数量--也被称为它的基数。 集合论中最早的问题之一是康托的连续统问题：确定实数线的基数。 在20世纪60年代，这个问题被证明是独立于通常的数学公理。 然而，目前还不清楚是否有其他令人信服的数学陈述，也是不可判定的，可能解决连续统问题。 摩尔的研究目的是建立新的分类结果的独立性，为无穷大结构，并将这些结果与连续问题。