Combinatorial Set Theory

摩尔在无限组合学中的工作涉及不可数结构的分类和基问题以及它们与基数幂的联系。一方面，强公理，如适当的强制公理和伍丁的Pmax公理被调用来建立嵌入和形态之间的结构，如线性顺序和拓扑空间。由于这些强公理本身暗示实线的基数是第二个不可数基数，因此很自然地要问，从它们推导出来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，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是关于无限集的“大小”的严格发展。基数提供了一个线性尺度，人们可以用它来衡量一个集合所具有的元素的数量——也称为它的基数。集合论中最早的问题之一是康托的连续问题：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的。然而，是否有其他令人信服的、同样不可确定的数学命题可以解决连续统问题，目前尚不清楚。Moore的研究旨在建立无限数学结构的新分类结果的独立性，并将这些结果与连续统问题联系起来。

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.