CAREER: Covering with Derived Models

PI提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基数的内部模型。他将专注于来自描述性内模型理论的问题，并将致力于推进hod小鼠理论。该项目的长期目标是证明鼠标集猜想，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提升到现有水平之外，并在可测量基数上计算出更好的方形失效一致性下界。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，正如科学成就的有效性必须经过检验一样，数学发现的一致性也必须经过检验。数学理论是建立在Zermelo-Frankel集合理论公理和选择公理（ZFC）的基础上的。如果ZFC及其各种扩展的证明是正确的，则可以认为得到的理论是正确的。集合论是通过大基数公理（Large Cardinal Axioms， LCA）处理ZFC及其扩展的一致性的数学部分。由于著名的哥德尔不完备定理，人们不能指望证明ZFC或它的任何扩展是一致的。然而，人们可以希望为ZFC + LCA提供自然发生的模型，就像自然数集是皮亚诺算术公理（PA）的自然模型一样。PA是指导和控制日常生活中算术使用的公理系统。内模程序是一个集合论程序，其主要目标是为ZFC + LCA构造这样的规范模型。提议的项目是对内部模型程序的贡献。提出的项目的一个具体目标是将最近最成功的用于构建ZFC的各种扩展的自然模型的方法，即核心模型归纳，提升到新的水平，从而构建以前无法达到的ZFC扩展的自然模型。

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.