Forcing and large cardinals

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The overall 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. Some of these axioms are purposely applicable outside set theory; others are of a nature that is, at face value, largely internal to set theory, but turn out to have effects on basic mathematical objects, for example on sets of real numbers. This project deals with axioms of both types, and with methods that compare their relative strengths. It involves the development of new axioms of the first type that should have applications in contexts that were previously out of reach, the construction of minimal models for axioms of the second type within set theory, and applications of both the axioms and their minimal models, within set theory and to the real numbers.This project deals with several central areas in set theory: (i) forcing axioms and their applications; (ii) inner models theory; (iii) applications of inner models theory to descriptive set theory; and (iv) 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. It is one of the goals of this project to develop these analogues further, and to use them in extending 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 construction, nature, and combinatorial properties of inner models at the level of supercompact cardinals. This is a long-standing project in the area and one that saw a great deal of recent progress. In connections with (iii) this project is concerned with applications of inner models theory at the level of Woodin cardinals to questions in descriptive set theory. The structure of inner models at this level is well understood, and there are well known connections to descriptive set theory. These connections already yielded solutions to several previously intractable questions in descriptive set theory. Finally, in connection with (iv) this project is primarily concerned with the tree property, a remnant of large cardinal strength that can consistently hold at small cardinals.