CAREER: Covering with Derived Models

职业：用派生模型进行覆盖

• 批准号: 1352034
• 负责人: Grigor Sargsyan
• 金额: $ 45万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-09-01 至 2020-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1352034&HistoricalAwards=false
• 关键词: CAREER; Covering; Derived; Models

The PI is proposing to work on various aspects of the inner model program, which is the program for constructing inner models for large cardinals. He will concentrate on problems coming from descriptive inner model theory and will work on advancing the theory of hod mice. The long term goal of the project is to prove the Mouse Set Conjecture, which is a central conjecture in descriptive inner model theory. Immediate goals include advancing the core model induction technique beyond its current levels, and computing better consistency lower bounds for the failure of square at a measurable cardinal. Mathematics is the language of science; it is what is used to express scientific predictions and theories. Mathematics is simply the most fundamental and indispensable part of science. However, just as scientific achievements must be tested for their validity, mathematical discoveries too must be tested for their consistency. Mathematical theories are based on axioms known as the axioms of Zermelo-Frankel set theory with the Axiom of Choice (ZFC). Provided that the proofs from ZFC and its various extensions are correct, it is believed that the resulting theories are correct. Set theory is the part of mathematics that deals with the consistency of ZFC and its extensions via Large Cardinal Axioms (LCA). Because of the celebrated Godel's incompleteness theorem one cannot hope to prove that ZFC or any of its extensions are consistent. Nevertheless, one can hope to provide naturally occurring models for ZFC + LCA in the same spirit that the set of natural numbers is the natural model of the axioms of Peano Arithmetic (PA). PA is the axiomatic system that guides and controls the usage of arithmetic in everyday life. The inner model program is a set theoretic program whose primary goal is to construct such canonical models for ZFC + LCA. The proposed project is a contribution to the inner model program. One specific objective of the proposed project is to advance the most successful recent method for constructing natural models for various extensions of ZFC, the core model induction, to new levels, and thus to construct natural models for extensions of ZFC that were unreachable before.

PI建议在内部模型程序的各个方面进行工作，该程序用于构建大型红衣主教的内部模型。他将专注于来自描述性内部模型理论的问题，并将致力于推进hod mice理论。该项目的长期目标是证明老鼠集猜想，这是描述性内模型理论中的一个中心猜想。近期目标包括将核心模型归纳技术推进到目前的水平之上，并计算出在可测量基数上平方失败的更好的一致性下限。数学是科学的语言;它是用来表达科学预测和理论的。数学是科学中最基本、最不可缺少的部分。然而，正如科学成就必须检验其有效性一样，数学发现也必须检验其一致性。数学理论是基于被称为Zermelo-Frankel集合论公理与选择公理（ZFC）的公理。只要ZFC及其各种扩展的证明是正确的，我们就相信所得到的理论是正确的。集合论是数学的一部分，通过大基数公理（LCA）处理ZFC及其扩展的一致性。由于著名的哥德尔不完备性定理，人们不能希望证明ZFC或其任何扩展是一致的。然而，人们可以希望为ZFC + LCA提供自然发生的模型，其精神与自然数集是皮亚诺算术（PA）公理的自然模型相同。PA是指导和控制日常生活中算术使用的公理系统。内部模型程序是一个集合论程序，其主要目标是为ZFC + LCA构建这样的规范模型。该项目是对内部模型计划的贡献。拟议项目的一个具体目标是推进最近最成功的方法，为ZFC的各种扩展构建自然模型，核心模型归纳，到新的水平，从而构建ZFC的扩展是以前无法达到的自然模型。