Combinatorial Set Theory, Forcing, and Large Cardinals

组合集合论、强迫和大基数

• 批准号: 1954117
• 负责人: Dima Sinapova
• 金额: $ 28万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-06-01 至 2023-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1954117&HistoricalAwards=false
• 关键词: Combinatorial; Set; Theory; Forcing; Large; Combinatorial; Set; Theory; Forcing; Large

Many natural questions in mathematics cannot be answered in terms of the standard axioms of mathematics alone. The most famous example is the Continuum Hypothesis, which states that any infinite subset of the real numbers is either countable or in a one-to-one correspondence with the whole set of real numbers. The technique known as forcing can be used to show that the Continuum Hypothesis is logically independent of the standard set of axioms for mathematics, known as Zermelo-Fraenkel set theory with the axiom of choice (ZFC). A major theme in modern set theory is development of relative consistency results and ZFC-strengthenings. ZFC-strengthenings are given by large cardinal hypotheses, which assert the existence of certain highly compact mathematical objects, called large cardinals, with strong reflection properties, that is, if a property holds at the large cardinal, it must hold at many cardinals below it. Assuming large cardinals exist, the method of forcing can be used to create various mathematical models. This project analyzes these constructions, motivated by two complementary notions: what is mathematically necessary? what is mathematically sufficient? The project provides research training opportunities for graduate students.The main objectives of the project are analyzing what is possible from large cardinals and forcing, versus what constraints are imposed by ZFC. Forcing in the presence of large cardinals is the main tool to create models of ZFC and prove consistency results. In contrast to having large cardinals, Gödel's constructible universe L is the minimal class model of set theory. Two central questions are how much the universe resembles L, and how much compactness can be achieved by forcing. Compactness is the phenomenon where if a certain property holds for every smaller substructure of a given object, then it holds for the object itself. It follows from large cardinals, and often fails in L. Compactness is captured by some key combinatorial principles such as the tree property and its strengthenings. On the other hand, square properties are the canonical instance of incompactness and serve as a yardstick how close a model is to L. The project will focus on the interplay between these principles and forcing extensions constructed from large cardinals. The PI will also investigate how they interact with cardinal arithmetic and especially with singular combinatorics.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.

仅凭数学的标准公理就不能回答数学中许多自然问题。最著名的例子是连续假设，该假设指出，实际数字的任何无限子集都是可计数的，要么与整个实数集合一对一。所谓强迫的技术可用于表明连续假设在逻辑上独立于数学的标准公理集，称为Zermelo-fraenkel集合理论，具有选择的公理（ZFC）。现代布景理论中的一个主要主题是相对一致性结果和ZFC增长的发展。 ZFC强度是由大型基本假设给出的，该假设断言某些高度紧凑的数学对象，称为大型红衣主教，具有强反射特性，也就是说，如果某个特性在大型的红衣主教处保持，则必须在其下方的许多红衣主教处保持。假设存在大型红衣主教，则可以使用强迫方法来创建各种数学模型。该项目分析了这些结构，这是由两个完整概念激励的：在数学上有什么需要？什么在数学上足够？该项目为研究生提供了研究培训机会。该项目的主要目标是分析了大型红衣主教和强迫的可能性，而ZFC施加了什么约束。在大型红衣主教的存在下强迫是创建ZFC模型并证明一致性结果的主要工具。与拥有大型的红衣主教相反，戈德尔的可构造宇宙是集合理论的最小类模型。两个核心问题是宇宙类似于L，以及通过强迫可以实现多少紧凑性。紧凑性是一种现象，如果某个特性适用于给定对象的每个较小的子结构，则它为对象本身保留。从大型红衣主教中开始，通常会失败。另一方面，正方形的性质是不兼容性的规范实例，并用作模型与L的尺寸。 PI还将调查他们如何与红衣主教算术相互作用，尤其是与单数组合学的相互作用。该奖项反映了NSF的法定任务，并使用基金会的知识分子优点和更广泛的影响审查标准，认为通过评估被认为是宝贵的支持。