Forcing and Large Cardinals

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The goal of this project is to develop a better understanding of the possible behaviors of the mathematical universe. Our knowledge of the mathematical universe comes through deduction from axioms. This knowledge is inherently incomplete, and there is a wide range of questions that cannot be decided from the standard axioms. Set theorists have developed and studied additional axioms that allow settling some of these questions. These axioms are broadly divided into two types: Large cardinal axioms, and forcing axioms. The large cardinal axioms form a natural hierarchy ordered by axiomatic strength. Forcing axioms have applications outside set theory. This project involves the development of new forcing axioms primarily meant to deal with questions related to the second uncountable cardinal, methods for applications of these axioms, the theory of large cardinal axioms primarily at the level of strength expected to be connected to the new forcing axioms (the level of supercompact cardinals), and other applications of large cardinal axioms at this strength, primarily to questions of infinitary combinatorics. While there has been a large body of work on forcing axioms related to the first uncountable cardinal, analogous forcing axioms related to the second uncountable cardinal were only discovered recently. Similarly the theory of large cardinal axioms at the level of supercompact cardinals is only now beginning to come into focus. This project will obtain results in these emerging subjects, develop methods that open these subjects to broader research, and aim to apply them to some long standing open problems.This project deals with several central areas in set theory: (i) forcing axioms and their applications; (ii) inner models theory; and (iii) infinitary combinatorics. Forcing axioms are strengthenings of the Baire category theorem that allow meeting a prescribed number of dense sets with filters in prescribed classes of partial orders. In connection with (i) this project is particularly concerned with higher analogues of the proper forcing axiom (PFA). PFA, developed in the early 1980s, allows meeting aleph_1 dense sets in proper partial orders. It has proved incredibly useful both as a starting point for consistency proofs and as an axiom leading to set theoretic structure theorems. Recent work of the PI shows that there are analogues of PFA which involve meeting more than aleph_1 dense sets. Separately, the PI developed new reflection principles at aleph_2. It is one of the goals of this project to combine the higher analogues of PFA with these reflection principles, and use the new resulting axioms to extend applications of PFA to new contexts. The inner models program has as its main goal the construction of models for large cardinal axioms from assumptions that do not directly involve large cardinals (for example from forcing axioms). In connection with (ii), this project is primarily concerned with the theory of inner models at the level of supercompact cardinals. In connections with (iii) this project is primarily concerned with the tree property, a remnant of large cardinal strength that can consistently hold at small cardinals. One of the goals of the project is to obtain it simultaneously at all successors in increasingly large intervals of cardinals.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.