Combinatorial Set Theory, Forcing, and Large Cardinals

Many natural questions in mathematics cannot be answered in terms of the standard axioms of mathematics alone. The most famous example is the Continuum Hypothesis, which states that any infinite subset of the real numbers is either countable or in a one-to-one correspondence with the whole set of real numbers. The technique known as forcing can be used to show that the Continuum Hypothesis is logically independent of the standard set of axioms for mathematics, known as Zermelo-Fraenkel set theory with the axiom of choice (ZFC). A major theme in modern set theory is development of relative consistency results and ZFC-strengthenings. ZFC-strengthenings are given by large cardinal hypotheses, which assert the existence of certain highly compact mathematical objects, called large cardinals, with strong reflection properties, that is, if a property holds at the large cardinal, it must hold at many cardinals below it. Assuming large cardinals exist, the method of forcing can be used to create various mathematical models. This project analyzes these constructions, motivated by two complementary notions: what is mathematically necessary? what is mathematically sufficient? The project provides research training opportunities for graduate students.The main objectives of the project are analyzing what is possible from large cardinals and forcing, versus what constraints are imposed by ZFC. Forcing in the presence of large cardinals is the main tool to create models of ZFC and prove consistency results. In contrast to having large cardinals, Gödel's constructible universe L is the minimal class model of set theory. Two central questions are how much the universe resembles L, and how much compactness can be achieved by forcing. 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. It follows from large cardinals, and often fails in L. Compactness is captured by some key combinatorial principles such as the tree property and its strengthenings. On the other hand, square properties are the canonical instance of incompactness and serve as a yardstick how close a model is to L. The project will focus on the interplay between these principles and forcing extensions constructed from large cardinals. The PI will also investigate how they interact with cardinal arithmetic and especially with singular combinatorics.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.