Forcing and Large Cardinals

强迫和大红衣主教

• 批准号: 1764029
• 负责人: Itay Neeman
• 金额: $ 50万
• 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: 2018
• 资助国家: 美国
• 起止时间: 2018-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1764029&HistoricalAwards=false
• 关键词: Forcing; Large; Cardinals

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.

该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。这些公理大致分为两类：大基数公理和强迫公理。大基数公理形成了一个由公理强度排序的自然层次结构。强迫公理在集合论之外也有应用。这个项目涉及新的强制公理的发展，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，主要是在强度水平上的大基数公理理论，预计将与新的强制公理相联系。（超紧基数的水平），和其他应用的大基数公理在这个强度，主要是无限组合问题。虽然已经有大量的工作，迫使公理有关的第一个不可数基数，类似的迫使公理有关的第二个不可数基数是最近才发现的。类似地，超紧基数水平上的大基数公理理论现在才开始成为焦点。该项目将在这些新兴学科中取得成果，开发方法，使这些学科能够进行更广泛的研究，并旨在将其应用于一些长期存在的开放问题。该项目涉及集合论的几个中心领域：（i）强迫公理及其应用;（ii）内部模型理论;和（iii）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是在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稠密集。另外，PI在aleph_2开发了新的反射原理。这是这个项目的目标之一，结合联合收割机的PFA与这些反射原则的更高的类似物，并使用新的结果公理扩展PFA的应用到新的上下文。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），本项目主要关注超紧基数水平上的内部模型理论。在连接（iii）这个项目主要是与树的财产，剩余的大基数的力量，可以始终保持在小基数。该项目的目标之一是同时获得它在所有继任者在越来越大的间隔红衣主教。这个奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

项目成果

Chang’s Conjecture with $$\square _{\omega _1, 2}$$ from an $$\omega _1$$-Erdős cardinal

张猜想 $$square _{omega _1, 2}$$ 来自 $$omega _1$$-ErdÅs 基数

• DOI: 10.1007/s00153-020-00723-w
• 发表时间: 2020
• 期刊: Archive for Mathematical Logic
• 影响因子: 0.3
• 作者: Neeman, Itay;Susice, John
• 通讯作者: Susice, John

THE TREE PROPERTY AT THE TWO IMMEDIATE SUCCESSORS OF A SINGULAR CARDINAL

单一红衣主教的两个直接继承人的树属性

• DOI: 10.1017/jsl.2020.11
• 发表时间: 2021
• 期刊: The Journal of Symbolic Logic
• 作者: CUMMINGS, JAMES;HAYUT, YAIR;MAGIDOR, MENACHEM;NEEMAN, ITAY;SINAPOVA, DIMA;UNGER, SPENCER
• 通讯作者: UNGER, SPENCER

On the powersets of singular cardinals in HOD

关于 HOD 中奇异基数的幂集

• DOI: 10.1090/proc/14913
• 发表时间: 2020
• 期刊: Proceedings of the American Mathematical Society
• 影响因子: 1
• 作者: Ben-Neria, Omer;Gitik, Moti;Neeman, Itay;Unger, Spencer
• 通讯作者: Unger, Spencer

The ineffable tree property and failure of the singular cardinals hypothesis

不可言喻的树性质和奇异基数假说的失败

• DOI: 10.1090/tran/8110
• 发表时间: 2020
• 期刊: Transactions of the American Mathematical Society
• 影响因子: 1.3
• 作者: Cummings, James;Hayut, Yair;Magidor, Menachem;Neeman, Itay;Sinapova, Dima;Unger, Spencer
• 通讯作者: Unger, Spencer

Abraham–Rubin–Shelah open colorings and a large continuum

亚伯拉罕·鲁宾·谢拉开放色彩和大连续体

• DOI: 10.1142/s0219061321500276
• 发表时间: 2022
• 期刊: Journal of Mathematical Logic
• 影响因子: 0.9
• 作者: Gilton, Thomas;Neeman, Itay
• 通讯作者: Neeman, Itay