Descriptive Inner Model Theory and Its Applications

描述性内模型理论及其应用

• 批准号: 1855757
• 负责人: Nam Trang
• 金额: $ 12.06万
• 依托单位: University of North Texas
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-09-01 至 2020-03-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1855757&HistoricalAwards=false
• 关键词: Descriptive; Inner; Model; Theory; Its

The Zermelo-Fraenkel axioms plus the Axiom of Choice (ZFC) have been widely accepted as a foundation for mathematics; virtually all branches of mathematics that have been studied and applied to other scientific fields can be interpreted in ZFC. However, it turns out that there are natural and important mathematical theories which cannot be decided by ZFC alone. The Large Cardinal Axioms (LCAs) are extensions of ZFC designed to settle all such theories. This is the Godel's program in Set Theory. If an LCA is "correct", then the theories it decides are also correct. How can one test for correctness of an LCA? The Inner Model Program, a major program in modern Set Theory, justifies correctness by constructing canonical models for LCAs much like the natural numbers are the canonical model for the Peano Axioms of Arithmetic (PA) (and as such PA is a correct theory). The proposed project contributes to the Inner Model Program by advancing methods for constructing canonical models for LCAs from various extensions of ZFC. The project focuses on studying the connections between inner models, sets of reals, hybrid structures (such as hereditarily ordinal definable sets (HOD) of determinacy models), forcing, and strong combinatorial principles (such as the Proper Forcing Axiom (PFA)). The PI proposes to work on advancing the basic theory of hybrid structures, focusing on strategy mice and hod mice as well as developing methods for the core model induction beyond its current state. In particular, the project aims to make advancements in answering the following fundamental questions in descriptive inner model theory: (1) Is HOD of a determinacy model fine-structural (e.g. do the Generalized Continuum Hypothesis (GCH), various square principles hold in HOD)? (2) What is the consistency strength of PFA? (3) Does PFA (or any other strong combinatorial theory) imply models in various partially backgrounded constructions iterable?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.

Zermelo-Fraenkel公理和选择公理（Axiom of Choice，ZFC）被广泛接受为数学的基础;几乎所有被研究和应用于其他科学领域的数学分支都可以在ZFC中解释。然而，事实证明，有自然的和重要的数学理论，不能决定由ZFC单独。大基数公理（LCAs）是ZFC的扩展，旨在解决所有这些理论。这是集合论中的哥德尔程序。如果LCA是“正确的”，那么它所决定的理论也是正确的。如何测试LCA的正确性？内模型程序是现代集合论中的一个主要程序，它通过为LCA构建规范模型来证明正确性，就像自然数是皮亚诺算术公理（PA）的规范模型一样（因此PA是正确的理论）。拟议的项目有助于内部模型计划的推进方法，从ZFC的各种扩展构建规范模型的LCA。该项目的重点是研究内部模型，实数集，混合结构（如确定性模型的遗传有序可定义集（HOD）），强迫和强组合原理（如正确强迫公理（PFA））之间的联系。PI建议致力于推进杂交结构的基础理论，重点关注策略小鼠和hod小鼠，以及开发超越当前状态的核心模型诱导方法。特别是，该项目旨在回答描述性内模型理论中的以下基本问题：（1）确定性模型的HOD是精细结构的吗（例如，广义连续统假设（GCH），各种平方原理在HOD中成立）？(2)PFA的稠度强度是多少？(3)PFA（或任何其他强组合理论）是否意味着模型在各种部分背景结构中是可迭代的？该奖项反映了NSF的法定使命，并被认为是值得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的支持。

项目成果

On Supercompactness of \omega_1

论omega_1的超紧性

• 发表时间: 2020
• 期刊: Springer Proceedings in Mathematics & Statistics
• 作者: Ikegami, Daisuke;Trang, Nam
• 通讯作者: Trang, Nam

Supercompactness Can Be Equiconsistent with Measurability

超紧凑性可以与可测量性等同

• DOI: 10.1215/00294527-2021-0031
• 发表时间: 2021
• 期刊: Notre Dame Journal of Formal Logic
• 影响因子: 0.7
• 作者: Trang, Nam