Descriptive Inner Model Theory

内模理论的主要开放问题是构造以超紧基数为主要目标的具有大基数的规范内模。60年代和70年代，在解决这一问题方面取得了重大进展。特别地，对于可测基数和强基数区域中的许多大基数，构造了这样的正则模型。然而，很快就发现了新的障碍，通过Martin、Steel和Woodin所做的基本工作，越来越明显的是，问题的最终解决方案必须纳入描述性集合理论的思想。80年代末90年代初，马丁、斯蒂尔和伍丁发现了内模理论和描述集合论之间的许多桥梁，从而统一了集合论的两个领域。描述性内模理论是通过他们的工作而产生的理论。它的主要技术问题是内部模型问题的描述性版本，即构造捕捉真理的规范模型的问题。虽然这个问题有许多正式的版本，但最广为人知的是鼠标集猜想(Mouse Set Conjecture，MSC)，它猜想在确定性模型中，序数可定义性这一最强大的可定义性形式可以通过集合论的规范模型来捕捉。Sargsyan在他的论文中，在Woodin早期工作的基础上，发展了证明MSC的技术，并使用这些技术获得了关于MSC的一些部分结果。这些部分结果随后被翻译成内部模型理论的普通语言，产生了具有大基数的规范模型。Sargsyan认为，对内部模型问题的描述方法是解决50年来内部模型问题的最有希望的途径，他计划进一步探索这一问题。集合论是数学的语言。它提供了基本的基础，其余的数学都建立在这些基础上。数学基础中最基本的问题之一是将所有的数学归结为一套基本的公理，这些公理的一致性要么可以被证明，要么可以令人信服地论证。哥德尔著名的不完全性定理暗示，第一种选择是不可能的。然而，由于上个世纪的重大发展，第二种选择可能通过一类被称为大型基数公理的无穷大公理来实现，该公理断言存在非常大的集合。实际上，正如哥德尔自己预测的那样，每一种已知的自然数学理论都被简化为某种大型的基本公理。21世纪的集合论者继承了论证这种公理的一致性的问题，这通常是通过展示典型的例子，或者更确切地说，这种公理的模型来完成的。展示这种大基数的正则模型的问题被称为内部模型问题，这是所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.