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.

该项目的目的是更好地理解数学宇宙的可能行为。我们对数学宇宙的了解是通过从公理中扣除的。这些知识本质上是不完整的，并且从标准公理中无法确定许多问题。设定理论家已经开发并研究了其他公理，可以解决其中一些问题。这些公理大致分为两种类型：大型基本公理和迫使公理。大型基本公理会形成一个由公理强度排序的天然层次结构。强迫公理在集合理论之外具有应用。该项目涉及开发新的强迫公理，主要是为了处理与第二个不可数的红衣主教，这些公理的应用方法有关的问题，这些公理的应用，大型基本公理的理论主要是在预期与新的强迫公理相关的强度水平上，与新的强度构成了这些问题的应用程序，以及其他问题的应用程序，这些问题是在这些问题上的应用。尽管在强迫与第一个不可数的基本主教相关的公理方面进行了大量工作，但最近才发现与第二个不可数的基础主教相关的类似的公理。同样，在超级紧张的基数水平上，大型基本公理的理论才开始焦点。该项目将在这些新兴主题中获得结果，开发通过更广泛的研究开放这些主题的方法，并旨在将其应用于一些长期存在的开放问题。该项目涉及设定理论中的几个中心领域：（i）强迫Axioms及其应用； （ii）内部模型理论；和（iii）无限制组合学。强迫公理是Baire类别定理的增强，它允许在规定的部分订单类别中使用有过滤器来满足规定数量的密集集。与（i）有关，该项目特别与适当强迫公理（PFA）的更高类似物有关。 PFA于1980年代初期开发，允许在适当的部分订单中遇到Aleph_1密集的集合。事实证明，这是一致性证明的起点，也是导致设定理论结构定理的公理的起点。 PI的最新工作表明，PFA的类似物涉及比Aleph_1密集的集合更多。另外，PI在Aleph_2开发了新的反射原理。它是该项目的目标之一，将PFA的更高类似物与这些反思原则相结合，并使用新的结果公理将PFA的应用扩展到新环境。内部模型程序作为其主要目标是从不直接涉及大型红衣主教的假设（例如强迫公理）的假设中建造大型基本公理的模型。与（ii）有关，该项目主要涉及超级紧张枢机主教的内部模型理论。在与（iii）的连接中，该项目主要与树木特性有关，这是一个可以始终如一地保持小枢机主教的大型基本强度的残余。该项目的目标之一是在所有越来越大的红衣主教间隔内同时在所有继任者中同时获得它。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛的审查标准通过评估来获得支持的。

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

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

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