Forcing and large cardinals

这个项目的总体目标是更好地理解数学世界的可能行为。我们对数学宇宙的知识来自于公理的演绎。这种知识本质上是不完整的，而且有许多问题不是标准公理所能决定的。集合论者已经开发和研究了其他公理，这些公理可以解决其中的一些问题。其中一些公理是有意应用于集合论之外的；另一些公理的本质是，从表面上看，很大程度上是集合论的内部公理，但事实证明，这些公理对基本的数学对象有影响，例如对实数集。这个项目涉及这两种类型的公理，以及比较它们相对优势的方法。它涉及到第一类新公理的发展，这些公理应该在以前无法达到的背景下应用，集合论中第二类公理的极小模型的构建，以及公理及其极小模型在集合论和实数中的应用。这个项目涉及集合论的几个中心领域：(I)强制公理及其应用；(Ii)内模型理论；(Iii)内模型理论在描述集合论中的应用；以及(Iv)无限组合学。强迫公理是Baire范畴定理的加强，它允许满足指定数量的稠密集和指定类别的偏序滤子。关于(I)，这个项目特别关注真强迫公理(PFA)的更高层次的类似。PFA是在20世纪80年代初发展起来的，它允许在适当的偏序下满足$\Aleph_1$稠密集。它已经被证明是非常有用的，既是一致性证明的起点，也是导致集合论结构定理的公理。PI最近的工作表明，存在PFA的类似物，它们涉及到满足$\Aleph_1$稠密集。本项目的目标之一是进一步开发这些类似物，并将它们用于将PFA的应用扩展到新的背景下。内部模型计划的主要目标是根据不直接涉及大基数的假设(例如，通过强制公理)构建大型基数公理的模型。关于(Ii)，这个项目主要涉及超紧基数水平上的内部模型的构造、性质和组合性质。这是该地区的一个长期项目，最近也取得了很大进展。关于(Iii)，这个项目涉及在Woodin基数水平上的内模理论在描述集合论问题上的应用。这一层次的内部模型的结构是很好理解的，并且与描述性集合论有众所周知的联系。这些联系已经解决了描述性集合论中几个以前难以解决的问题。最后，关于(Iv)，这个项目主要涉及树的属性，这是一种大基数力量的残余物，可以持续保持在小基数上。

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.