Combinatorial Set Theory

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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.