Descriptive Inner Model Theory

集合论是一个相对较新的数学分支，它起源于数学分析和哲学。它的哲学开端与数学的基础有关，而数学在19世纪末和20世纪初并没有坚实的基础。当时的集合理论家设计了一个公理系统，称为Zermelo-Fraenkel与选择公理（ZFC），消除了当时的所有悖论，并将所有数学置于更坚实的基础上。另一方面，康托，工作的问题来自研究傅立叶序列，需要设备较长的归纳然后一组自然数让他这样做。他的工作使他什么是现在被称为基数，序数和以上所有的研究的真实的数字。他的工作导致了所谓的连续统假设，这是希尔伯特臭名昭著的问题清单上的第一个开放问题。哥德尔和科恩最终证明了连续统假设是不可判定的，即既不能从ZFC证明也不能从ZFC证明，从而暴露了公理系统ZFC的一个主要弱点。它不决定关于实数集的简单问题，实数集是最基本的数学对象之一。描述性内模型理论是现在被称为哥德尔计划的一部分，哥德尔计划是通过研究越来越强的自然公理系统来消除数学中的不可判定性，扩展ZFC，决定越来越多关于实数集和其他数学对象的自然问题。哥德尔本人建议，大枢机主教阶层应该用于上述目的。描述性内模型理论的目标是建立大基数的规范模型，并研究它们对真实的数集的影响。例如，由于许多作者的最大成就之一是，给定大基数，实数的投射集的行为完全是实数的Borel集，就其正则性而言（即可测性，范畴等）。此外，该项目还为研究生提供研究培训机会。PI建议解决该领域的一些核心开放问题，如HOD问题或迭代问题。HOD问题本质上是说，在实数集满足人们所期望的所有正则性的模型中，由序数可遗传定义的集合组成的宇宙是大基数的正则内模型。它连接理想的正则性属性的集合的reals与存在的大基数在一个根本的方式表明，正则性属性的可定义集合的reals是一个后果的大基数，反之亦然。HOD问题在90年代末和21世纪初被孤立，成为该地区最核心的开放问题之一。迭代性问题具有类似的性质，但避免了确定性模型。该奖项反映了NSF的法定使命，并被认为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来支持。

Set theory is a relatively new branch of mathematics that has its roots in mathematical analysis and philosophy. Its philosophical beginning had to do with the foundations of mathematics, which at the end of the 19th century and at the beginning of the 20th century was not standing on firm grounds. Set theorists of the time devised an axiomatic system known as Zermelo-Fraenkel with the Axiom of Choice (ZFC), that removed all paradoxes of the time, and put all of mathematics on firmer grounds. On the other hand, Cantor, working on questions coming from the study of Fourier sequences, needed to device longer inductions then the set of natural numbers allowed him to do. His work led him to what is now known as cardinal numbers, ordinal numbers and above all the study of the real numbers. His work led to what is called The Continuum Hypothesis, which was the first open problem on Hilbert's infamous list of problems. Godel and Cohen eventually showed that The Continuum Hypothesis is undecidable, i.e. neither provable nor disprovable, from, ZFC, thus exposing a major weakness in the axiomatic system ZFC. It does not decide simple questions about sets of reals, one of the most basic mathematical objects. Descriptive Inner Model Theory is one part of what is now called Godel's Program, which is the program of removing undecidability from mathematics by studying stronger and stronger natural axiomatic systems extending ZFC that decide more and more natural questions about sets of reals and other mathematical objects. Godel himself suggested that the Large Cardinal Hierarchy should be used for the aforementioned purpose. The goal of descriptive inner model theory is to build canonical models for large cardinals and also study their impact on the set of real numbers. For example, one of the greatest achievements of the subject due to many authors is that given large cardinals, projective sets of reals behave exactly as Borel sets of reals as far as their regularity properties are concerned (i.e. measurability, category and etc). In addition the project also provides research training opportunities for graduate students.The PI is proposing to attack some of the central open problems of the area such as the HOD Problem or the Iterability Problem. The HOD Problem essentially says that in models where sets of reals posses all regularity properties one may desire, the universe consisting of sets that are hereditarily definable from ordinals is a canonical inner model for large cardinals. It connects the desirable regularity properties of sets of reals with the existence of large cardinals in a fundamental way showing that the regularity properties of definable sets of reals are a consequence of large cardinals and vice versa. The HOD Problem was isolated in late 90s and early 2000s as one of the most central open problems of the area. Iterability Problem is of similar nature but avoids determinacy models.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.