Combinatorial Set Theory

Cornell University, Ithaca NY

Investigators

Abstract

Moore's work in infinite combinatorics concerns classification and basis problems for uncountable structures and their connection with cardinal exponentiation. On one hand, strong axioms such as the Proper Forcing Axiom and Woodin's Pmax axiom are invoked to build embeddings and morphisms between structures such as linear orders and topological spaces. Since these strong axioms themselves imply that the cardinality of the real line is the second uncountable cardinal, it is natural to ask whether the classification theorems which follow from them already fix the cardinality of the continuum. The research supported by this grant aims to prove new classification results from these strong axioms, to better understand the relationship between these classification results and the value of the continuum, and to improve our understanding of 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 scale with which one can measure the number of elements a set has --- also known as its cardinality. One of the earliest questions in set theory was Cantor's Continuum Problem: Determine the cardinality of the real number line. In the 1960s, this problem was shown to be independent of the usual axioms of mathematics. Still, it is unclear whether some other compelling mathematical statement, also undecidable, might settle the Continuum Problem. Moore's research aims to both establish the independence of new classification results for infinite mathematical structures and to relate these results to the Continuum Problem.