Infinitary Combinatorics

Set theory is the study of the foundations of mathematics. Many natural mathematical questions are independent of the usual mathematical axioms. The most famous example is the continuum hypothesis (CH), which is the statement that any infinite set of real numbers is either countable or has the same size as all the reals. This became Hilbert's First Problem. The first breakthrough was in 1940 by Kurt Godel, who showed that CH cannot be refuted by the standard mathematical axioms, known as ZFC (Zermelo-Fraenkel set theory with the axiom of choice). Then in 1964 Paul Cohen invented the method of forcing and used it to show that the continuum hypothesis is actually independent of ZFC. In other words, neither CH, not its negation is a logical consequence of the ZFC axioms. Since Cohen's work, modern set theory investigates ZFC constraints (i.e. "what is necessary") versus relative consistency results obtained by forcing (i.e. "what is possible"). Both questions are addressed by infinitary combinatorics, the study of infinite objects in mathematics. This study generates many projects and research training opportunities for graduate students.The PI will focus on using forcing and large cardinals to investigate properties of infinite objects. The main motivation is analyzing ZFC-constraints against consistency results. The project will center on analyzing combinatorial principles, such as the tree property, stationary reflection and square principles, and their relation to cardinal arithmetic, especially singular cardinal arithmetic. The tree property and stationary reflection are compactness type properties that follow from large cardinals. Compactness is the phenomenon where if a certain property holds for every smaller substructure of a given object, then it holds for the object itself. On the other hand, square properties are canonical instances of incompacntess that hold in Godel's constructible universe L and are at odds with large cardinals. This project will analyze their interplay and interaction with cardinal arithmetic.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.