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。我们对数学世界的认识来自于对公理的演绎。这种知识本质上是不完整的，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其他公理来解决这些问题。这些公理大致分为两类：大基数公理和强制公理。大的基本公理形成了按公理强度排序的自然等级。强迫公理在集合论之外也有应用。该项目涉及发展新的强制公理，主要用于处理与第二不可数基数相关的问题，这些公理的应用方法，主要在强度水平上与新强制公理（超紧基数水平）相关联的大基数公理理论，以及在此强度下大基数公理的其他应用，主要用于无穷大组合问题。虽然与第一个不可数基数相关的强制公理已经有了大量的工作，但与第二个不可数基数相关的类似强制公理直到最近才被发现。类似地，在超紧基数水平上的大基数公理理论现在才开始成为焦点。本项目将获得这些新兴学科的成果，开发使这些学科得到更广泛研究的方法，并旨在将其应用于一些长期存在的开放性问题。本项目涉及集合论中的几个中心领域：(i)强迫公理及其应用；（ii）内模理论；（三）无穷组合。强迫公理是对贝尔范畴定理的强化，它允许在规定的偏序类中满足规定数量的密集集和过滤器。在(i)方面，本项目特别关注适当强迫公理（PFA）的高级类似物。PFA是在20世纪80年代初发展起来的，它允许满足适当偏序的alph_1密集集。它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，PFA的类似物涉及满足多于α _1密集集。另外，PI在alph_2处发展了新的反射原理。这个项目的目标之一是将PFA的高级类似物与这些反射原理结合起来，并使用新的结果公理将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。与（ii）相联系，本项目主要关注超紧基数级别的内模型理论。在与（iii）的联系中，该项目主要关注树的属性，这是大基数强度的残余，可以始终保持小基数。该项目的目标之一是在所有继任者中以越来越大的基数间隔同时获得它。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.