Forcing, inner models, and large cardinals.

这个项目旨在帮助我们理解数学宇宙的结构特性。我们的数学知识来自于公理的演绎。这种知识本质上是不完整的，因为有很多问题永远无法从标准公理中得到答案，无论是肯定的还是否定的。集合理论家研究这些问题，看看它们是如何相互联系的，以及它们是如何与附加公理的主干联系起来的，这些公理被称为大基数公理，它们断言非常大的集合具有强反射特性。本项目通过三个相互关联的方面来探讨其中的一些问题。第一个问题涉及基数的一个性质，这个性质可以看作是一个大基数公理的残余。自20世纪70年代以来，人们一直在研究这一性质，对它的研究推动了一致性证明领域的实质性发展。第二个涉及到实数线的组合性质和实数集的大小之间的相互作用。在这两个方面，证明相关原理可以成立使用大的基本公理。第三个方面是研究大基本公理本身，为这些公理建立具体的模型，并进一步加强一些公理与实数线性质之间已知的联系。在所有情况下，总的目的都是进一步了解数学世界中什么是可能的，什么是不可能的。该项目将支持加州大学洛杉矶分校数理逻辑研究生的发展和培训。此外，本项目建立在PI以前在本科阶段的工作基础上。PI计划帮助加州大学洛杉矶分校有才华的本科生获得数学逻辑方面的研究生水平知识，并继续研究生学习。本项目涉及集合论中三个相互关联的领域：(i)树的性质；（ii）大连续体的一致性证明；（iii）大型枢机和内部模型，在伍丁枢机和超紧凑枢机区域。与(i)相关，该项目特别关注在越来越大的间隔中，在第一个不可数基数之上的常规基数上强制树属性。最终的目标是查看这些基数上的tree属性是否可以阻止该属性在其他基数上保持不变，或者tree属性是否一致地在所有这些基数上保持不变。我们离这个问题的答案还有很长的路要走，但近年来已经取得了一些令人印象深刻的进展，这个项目试图在这一进展的基础上进一步突破界限。在（ii）中，该项目特别关注适当强迫公理（PFA）的一些中心结构结果，例如Todorcevic的开放着色公理和p-理想二分法。该项目旨在确定这些原则是否与大连续体一致，即比PFA下的尺寸更大。最终目标是开发具有大连续体的一致性结果的健壮框架，或者确定此类框架的实际数学障碍。与（iii）相关的是，该项目特别关注在Woodin基数水平上的内模型理论的描述性集合理论应用，并将内模型理论推向超紧基数水平。前者应该导致解决一些更棘手的，仍然开放的问题，在确定性下的描述性集合理论。后者应该有助于我们理解大基数层次的一个关键层次。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。

This project seeks to contribute to our understanding of structural properties of the universe of mathematics. Our knowledge of mathematics comes through deduction from axioms. This knowledge is inherently incomplete, in the sense that there is a wide range of questions that can never be answered, either positively or negatively, from the standard axioms. Set theorists study these questions, to see how they relate to each other, and how they relate to a backbone of additional axioms, called large cardinal axioms, that assert strong reflection properties for very large sets. This project explores some of these questions through three interrelated aspects. The first involves a property of cardinal numbers that can be viewed as a remnant of a large cardinal axiom. This property has been studied since the 1970s, and work on it has been driving substantial developments in the area of consistency proofs. The second involves the interactions between combinatorial properties of the line of real numbers, and the size of the set of real numbers. In both these aspects, proofs that the relevant principles can hold use large cardinal axioms. The third aspect is to study the large cardinal axioms themselves, to develop specific models for these axioms, and to further strengthen the known connection between some of these axioms and properties of the line of real numbers. The overall purpose in all cases is to further our understanding of what is, and what is not, possible in the universe of mathematics. This project will support the development and training of graduate students in mathematical logic at UCLA. In addition this project builds on previous work of the PI at the undergraduate level. The PI plans to help talented UCLA undergraduates acquire graduate level knowledge in mathematical logic, and continue to graduate studies. This project deals with three interrelated areas in set theory: (i) the tree property; (ii) consistency proofs with large continuum; and (iii) large cardinals and inner models, in the region of Woodin cardinals and of supercompact cardinals. In connection with (i), the project is particularly concerned with forcing the tree property at regular cardinals, above the first uncountable cardinal, in increasingly large intervals. The ultimate goal is to see whether the tree property on some of these cardinals can prevent the property from holding on other cardinals, or whether it is consistent that the tree property holds at all these cardinals. We are very far from an answer to this question, but there has been some impressive progress in recent years and this project seeks to build on this progress in order to push the boundaries further. In connection with (ii), the project is particularly concerned with some of the central structural consequences of the proper forcing axiom (PFA), for example Todorcevic's open coloring axiom and p-ideal dichotomy. This project seeks to determine whether these principles are consistent with large continuum, meaning larger than its size under PFA. The ultimate goal is to develop a robust framework of consistency results with large continuum, or determine actual mathematical obstacles for such frameworks. In connection with (iii), the project is particularly concerned with descriptive set theoretic applications of inner models theory at the level of Woodin cardinals, and with pushing the theory of inner models to the level of supercompact cardinals. The former should lead to solutions for some of the more intractable, still open, questions in descriptive set theory under determinacy. The latter should help with our understanding of a key level of the large cardinal hierarchy.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.