Forcing and large cardinals

该项目的总体目标是更好地理解数学宇宙的可能行为。我们对数学世界的认识来自于公理的演绎。这种知识本质上是不完整的，并且有很多问题不能从标准公理中决定。集合理论家已经发展和研究了额外的公理，允许解决这些问题。这些公理中有一些是有意地应用于集合论之外的;另一些则具有表面上看在很大程度上是集合论内部的性质，但结果证明对基本数学对象有影响，例如对真实的数的集合。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。它包括第一类新公理的发展，这些公理应该在以前无法达到的情况下得到应用，在集合论中为第二类公理构造最小模型，以及公理及其最小模型在集合论中和对真实的数的应用。（ii）内部模型理论;（iii）内部模型理论在描述集合论中的应用;（iv）无穷组合学。强迫公理是Baire范畴定理的加强，它允许满足规定数量的稠密集与规定类别的偏序中的过滤器。关于（i），本项目特别关注适当强迫公理（PFA）的更高类似物。PFA，在20世纪80年代初开发的，允许满足$\aleph_1 $密集集在适当的偏序。它已被证明是非常有用的，无论是作为一致性证明的起点，还是作为导致集合论结构定理的公理。PI最近的工作表明，有类似的PFA涉及会议超过$\aleph_1 $密集集。本项目的目标之一是进一步开发这些类似物，并将其用于将PFA的应用扩展到新的环境中。内部模型程序的主要目标是从不直接涉及大基数的假设（例如从强制公理）中构建大基数公理的模型。关于（ii），这个项目主要关注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，最近取得了很大进展。在连接（iii）这个项目是关注的应用程序的内部模型理论的水平上的Woodin基数的问题，在描述集理论。这个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着众所周知的联系。这些联系已经解决了描述集合论中几个以前难以解决的问题。最后，关于（iv），这个项目主要关注树的属性，一个大基数强度的残余，可以始终保持在小基数上。

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.