CAREER: Current and Future Developments of the Core Model Induction
职业:核心模型归纳的当前和未来发展
基本信息
- 批准号:1945592
- 负责人:
- 金额:$ 40.37万
- 依托单位:
- 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:2020-09-01 至 2025-08-31
- 项目状态:未结题
- 来源:
- 关键词:
项目摘要
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 Gödel'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). This project contributes to the Inner Model Program by advancing methods for constructing canonical models for LCAs from various extensions of ZFC. The educational component includes annual conferences, curriculum development, and outreach and broadening participation activities. The project fits into the general framework of 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 problem of building a canonical inner model for an LCA is referred to as the Inner Model Problem for that LCA. One of the project's main goals is to identify certain obstructions that may prevent the current approaches to resolving the Inner Model Problem for various LCAs from going further as well as proposing ways of overcoming them. Specifically, the project will study obstacles to the current techniques of the Core Model Induction (one such obstacle is the Sealing of the Universally Baire Sets) and develop new approaches to the Core Model Induction that overcome these obstacles. One main application of the above analyses is in constructing canonical inner models of large cardinals from various strong extensions of ZFC, like the Proper Forcing Axiom.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。这是哥德尔的集合论程序如果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)。该项目通过推进从ZFC的各种扩展构建LCA的规范模型的方法,为内部模型计划做出贡献。教育部分包括年度会议、课程编制以及外联和扩大参与活动。该项目适合于研究内部模型,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))之间的联系的一般框架。为LCA构建规范内部模型的问题被称为该LCA的内部模型问题。该项目的主要目标之一是确定某些障碍,这些障碍可能会阻止当前解决各种LCA内部模型问题的方法进一步发展,并提出克服这些障碍的方法。具体而言,该项目将研究核心模型归纳的现有技术的障碍(其中一个障碍是通用Baire集的密封),并开发克服这些障碍的核心模型归纳的新方法。上述分析的一个主要应用是从ZFC的各种强扩展中构建大基数的规范内部模型,如适当的强迫公理。该奖项反映了NSF的法定使命,并通过使用基金会的智力价值和更广泛的影响审查标准进行评估,被认为值得支持。
项目成果
期刊论文数量(8)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
COUNTABLE LENGTH EVERYWHERE CLUB UNIFORMIZATION
随处可见的俱乐部统一长度
- DOI:10.1017/jsl.2022.78
- 发表时间:2022
- 期刊:
- 影响因子:0
- 作者:CHAN, WILLIAM;JACKSON, STEPHEN;TRANG, NAM
- 通讯作者:TRANG, NAM
More definable combinatorics of around the first and second uncountable cardinal
第一个和第二个不可数基数的更可定义的组合
- DOI:
- 发表时间:2023
- 期刊:
- 影响因子:0.9
- 作者:Chan, William;Jackson, Stephen;Trang, Nam
- 通讯作者:Trang, Nam
Almost Everywhere Behavior of Functions According to Partition Measures
几乎处处根据分区度量的函数行为
- DOI:10.1017/fms.2023.130
- 发表时间:2024
- 期刊:
- 影响因子:0
- 作者:Chan, William;Jackson, Stephen;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:
- 影响因子:0.7
- 作者:Trang, Nam
- 通讯作者:Trang, Nam
The exact consistency strength of the generic absoluteness for the universally Baire sets
普适贝尔集的泛绝对性的精确一致性强度
- DOI:10.1017/fms.2023.127
- 发表时间:2024
- 期刊:
- 影响因子:0
- 作者:Sargsyan, Grigor;Trang, Nam
- 通讯作者:Trang, Nam
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Nam Trang其他文献
Preservation of AD via forcings
- DOI:
10.1007/s11856-025-2767-5 - 发表时间:
2025-05-09 - 期刊:
- 影响因子:0.800
- 作者:
Daisuke Ikegami;Nam Trang - 通讯作者:
Nam Trang
$mathsf {Sealing}$ from iterability
$mathsf {密封}$ 的可迭代性
- DOI:
- 发表时间:
2021 - 期刊:
- 影响因子:0
- 作者:
G. Sargsyan;Nam Trang - 通讯作者:
Nam Trang
Derived Models , and Σ 1-Reflection
派生模型和 Σ 1-反射
- DOI:
- 发表时间:
2010 - 期刊:
- 影响因子:0
- 作者:
J. Steel;Nam Trang - 通讯作者:
Nam Trang
STRUCTURE THEORY OF L(ℝ, μ) AND ITS APPLICATIONS
L(ℝ,μ)的结构理论及其应用
- DOI:
10.1017/jsl.2014.65 - 发表时间:
2015 - 期刊:
- 影响因子:0
- 作者:
Nam Trang - 通讯作者:
Nam Trang
BSL volume 29 issue 2 Cover and Front matter
BSL 第 29 卷第 2 期封面和封面
- DOI:
10.1017/bsl.2023.20 - 发表时间:
2023 - 期刊:
- 影响因子:0
- 作者:
G. Bezhanishvili;S. Kuhlmann;K. Bimbó;Øystein Linnebo;P. Dybjer;A. Muscholl;A. Enayat;Arno Pauly;Albert Atserias;Antonio Montalbán;M. Atten;V. D. Paiva;Clinton Conley;Christian Retoré;D. Macpherson;Nam Trang;Sandra Müller - 通讯作者:
Sandra Müller
Nam Trang的其他文献
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{{ truncateString('Nam Trang', 18)}}的其他基金
Descriptive Inner Model Theory and Its Applications
描述性内模型理论及其应用
- 批准号:
1855757 - 财政年份:2019
- 资助金额:
$ 40.37万 - 项目类别:
Standard Grant
Descriptive Inner Model Theory, Large Cardinals, and Combinatorics
描述性内模型理论、大基数和组合学
- 批准号:
1849295 - 财政年份:2018
- 资助金额:
$ 40.37万 - 项目类别:
Standard Grant
Descriptive Inner Model Theory, Large Cardinals, and Combinatorics
描述性内模型理论、大基数和组合学
- 批准号:
1565808 - 财政年份:2016
- 资助金额:
$ 40.37万 - 项目类别:
Standard Grant
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