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，以及它们与基数求幂的关系。一方面，引用强公理，如真的力公理和Woodin的Pmax公理，在线性序和拓扑空间等结构之间建立嵌入和态射。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，自然会问，由此而来的分类定理是否已经确定了连续统的基数。这项资助的研究旨在证明这些强公理的新的分类结果，更好地理解这些分类结果与连续统的价值之间的关系，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的“大小”的严格发展。基数提供了一个线性标度，人们可以用它来测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体问题：确定实数直线的基数。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。尽管如此，目前还不清楚其他一些同样无法确定的令人信服的数学陈述是否会解决连续统问题。摩尔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.