Descriptive Inner Model Theory

内模理论的主要开放问题是以超紧基数为主要目标的大基数正则内模的构造。60年代和70年代，在解决这一问题方面取得了重大进展。特别是，在可测量基数和强基数的区域中，构建了许多大基数的规范模型。然而，很快就发现了新的障碍，通过马丁、斯蒂尔和伍丁所做的基础性工作，人们越来越清楚地认识到，这个问题的最终解决方案必须结合描述性集合论的思想。在80年代末和90年代初，Martin， Steel和Woodin发现了内在模型理论和描述性集合理论之间的许多桥梁，从而统一了集合理论的两个领域。描述性内模型理论是在他们的工作中出现的理论。它的主要技术问题是内模型问题的描述版本，即构建捕获真理的规范模型的问题。虽然这个问题的许多正式版本都是可用的，但最公开的一个是鼠标集猜想（MSC），它推测在确定性模型中，有序可定义性，最强大的可定义性形式，可以通过集合理论的规范模型来捕获。Sargsyan在他的论文中，以Woodin的早期工作为基础，开发了证明MSC的技术，并使用这些技术获得了MSC的一些部分结果。这些部分结果然后被翻译成内模型理论的普通语言，产生具有大基数的规范模型。Sargsyan认为，内部模型问题的描述性方法是解决已有50年历史的内部模型问题的最有希望的途径，他计划进一步研究。集合论是数学的语言。它提供了数学的其他部分赖以建立的基础。数学基础中最基本的问题之一是将所有数学归结为一组基本公理，这些公理的一致性要么可以被证明，要么可以被令人信服地论证。Gödel著名的不完备定理暗示第一种选择是不可能的。然而，由于上个世纪的重大发展，第二种选择是可能的，通过一类被称为大基数公理的无限公理，它断言存在非常大的集合。事实上，正如Gödel自己所预测的那样，每一个已知的自然数学理论都被简化为一些大的基本公理。21世纪的集合理论家继承了论证这些公理的一致性的问题，这通常是通过展示这些公理的典型例子或模型来完成的。对于大基数显示这样的规范模型的问题被称为内模型问题，这是提出的研究的主要主题。

The main open problem of inner model theory is the construction of canonical inner models with large cardinals with supercompact cardinals being the main target. In 60s and 70s, major progress was made towards the resolution of the problem. In particular, such canonical models were constructed for many large cardinals in the region of measurable cardinals and strong cardinals. However, soon new obstacles were discovered and through a fundamental work done by Martin, Steel and Woodin, it became increasingly clear that the ultimate resolution of the problem has to incorporate ideas from descriptive set theory. In late 80s and early 90s, Martin, Steel and Woodin, discovered many bridges between inner model theory and descriptive set theory thus unifying the two areas of set theory. Descriptive inner model theory is the theory that has emerged through their work. Its main technical problem is the descriptive version of the inner model problem, namely, the problem of constructing canonical models capturing the truth. While many formal versions of this problem are available, the one that has been publicized most is the Mouse Set Conjecture (MSC) which conjectures that in models of determinacy, ordinal definability, the most powerful form of definability, can be captured via canonical models of set theory. Sargsyan, in his thesis, building on an earlier work of Woodin, developed techniques for proving MSC and used these techniques to obtained some partial results on MSC. These partial results then were translated into the ordinary language of inner model theory, producing canonical models with large cardinals. Sargsyan believes that the descriptive approach to the inner model problem is the most promising route to the resolution of the 50 year old inner model problem and he plans to pursue it further.Set theory is the language of mathematics. It provides the basic foundations above which the rest of mathematics is build. One of the most basic problems in the foundations of mathematic has been the reduction of all of mathematics into basic set of axioms whose consistency can either be proven or convincingly argued for. Gödel's celebrated incompleteness theorems imply that the first option is impossible. However, because of major developments in the previous century, the second option is possible via a class of axioms of infinity known as large cardinal axioms which assert the existence of very large sets. Virtually, as Gödel himself predicted, every know natural mathematical theory has been reduced to some large cardinal axiom. The set theorists of the 21st century inherited the problem of arguing for the consistency of such axioms which is usually done by exhibiting canonical examples or rather models of such axioms. The problem of exhibiting such canonical models for large cardinals has been known as the inner model problem which is the main subject of the proposed research.