Forcing, inner models, and large cardinals.

强迫、内部模型和大基数。

• 批准号: 2246905
• 负责人: Itay Neeman
• 金额: $ 36万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2246905&HistoricalAwards=false
• 关键词: Forcing; inner; models; large; cardinals.

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.

这个项目旨在帮助我们理解数学宇宙的结构属性。我们的数学知识是从公理中演绎出来的。这种知识本质上是不完整的，从某种意义上说，有很多问题永远不能从标准公理中得到回答，无论是肯定的还是否定的。集合论者研究这些问题，以了解它们是如何相互联系的，以及它们如何与附加公理的骨干联系起来，这些公理被称为大型基数公理，这些公理断言对非常大的集合具有很强的反射性质。本项目通过三个相互关联的方面探讨其中一些问题。第一个涉及基数的一个性质，它可以被视为一个大的基数公理的残余。自20世纪70年代以来，人们一直在研究这一性质，有关它的工作一直在推动一致性证明领域的实质性发展。第二个涉及实数行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，使用大型基本公理来证明相关原理是成立的。第三个方面是研究大型基数公理本身，为这些公理开发特定的模型，并进一步加强其中一些公理与实数线的性质之间的已知联系。在所有情况下，总体目的都是为了加深我们对数学宇宙中什么是可能的，什么是不可能的理解。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，该项目建立在本科生PI以前的工作基础上。PI计划帮助加州大学洛杉矶分校有才华的本科生获得研究生水平的数理逻辑知识，并继续研究生学习。这个项目涉及集合论中三个相互关联的领域：(I)树的性质；(Ii)大连续统的一致性证明；(Iii)大基数和内部模型，在Woodin基数和超紧基数的区域。关于(I)，该项目特别关注以越来越大的间隔在第一个不可数基数之上的规则基数处强制树的性质。最终目标是看看这些基数中的一些基数的树属性是否可以防止该属性保留其他基数，或者树属性是否一致地保留所有这些基数。我们对这个问题的答案还很遥远，但近年来取得了一些令人印象深刻的进展，该项目试图在这一进展的基础上再接再厉，以进一步推动这一界限。关于(Ii)，该项目特别关注适当的强迫公理(PFA)的一些中心结构结果，例如托多尔切维奇的开放着色公理和p-理想二分法。本项目试图确定这些原则是否符合大的连续体，即大于PFA下的大小。最终目标是开发一个具有大连续体的结果一致性的健壮框架，或确定此类框架的实际数学障碍。关于(Iii)，本项目特别关注内模型理论在Woodin基数水平上的描述性集合论应用，并将内模型理论推向超紧基数水平。前者应该导致解决确定性下描述集合论中一些更棘手、仍然悬而未决的问题。后者应该有助于我们理解大型枢机主教等级的关键级别。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。