Forcing and Large Cardinals

这个项目的目标是更好地理解数学宇宙的可能行为。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。这些公理大致分为两类：大型基数公理和强迫公理。大的基数公理形成了一个按公理强度排序的自然层次。强迫公理在集合论之外也有应用。这个项目包括发展新的强迫公理，主要是为了处理与第二不可数基数有关的问题，这些公理的应用方法，主要是强度水平的大型基数公理理论，预计连接到新的强迫公理(超紧基数水平)，以及在这种强度下的大型基数公理的其他应用，主要是无限组合学的问题。虽然已经有大量关于与第一个不可数基数相关的强制公理的工作，但与第二个不可数基数相关的类似的强制公理直到最近才被发现。同样，超紧基数水平上的大型基数公理理论现在才开始成为焦点。这个项目将在这些新兴学科中取得成果，开发方法，使这些学科向更广泛的研究开放，并旨在将它们应用于一些长期悬而未决的问题。本项目涉及集合论的几个核心领域：(I)强迫公理及其应用；(Ii)内部模型理论；(Iii)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，它们涉及到超过Aleph_1稠密集的相遇。另外，PI在Aleph_2上开发了新的反射原理。本项目的目标之一是将PFA的高级类似物与这些反射原理相结合，并使用新的公理将PFA的应用扩展到新的背景。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理论。在与(Iii)的联系中，这个项目主要涉及树的属性，这是一种大基数力量的残余，可以持续地保持在小基数上。该项目的目标之一是在越来越大的红衣主教间隔内同时获得所有继任者的奖项。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.