Mathematical Sciences: Set Theory

弗里德曼打算研究集合论中的四个领域 除了开始写一本关于L编码方法的书之外。 (The后者应该有助于使许多重要的应用， 这种方法可以为更广泛的集合理论家所接受。） （一） 与他以前的学生Dorshka Wylie（现在是NSF的研究员， 麻省理工学院讲师），他正致力于一种全新的方法， 强基数的核心模型K，它应该提供一个 一个强有力的新工具，建立组合原理K。 (2)第二个目标是使编码方法能够平滑地提升到 K，允许扩展的结果，如他的证明的Pi-1 - 2- Singleton Conjecture到核心模型上下文。 (3)很长的- 迭代强迫关注理论中的一个悬而未决的问题 可数支集迭代法的推广问题 而不破坏红雀。 若干重要 通过这种技术可以解决一致性问题。 与他 学生迪米特里罗斯齐马斯，弗里德曼将试图开发这样一个 方法通过使用沼泽。 (4)弗里德曼还将探讨 描述集合论中的问题是， 射影集合的结构理论可以直接从 大基数的存在性，而不使用无限对策 理论 集合论提供了一种流行的方法， 所有的数学，最著名的系统化尝试， 罗素和怀特海的《数学原理》 20世纪早期。 重要的基础性 问题涉及的独立性和一致性， 用于建立集合论的公理。 策梅洛和弗伦克尔公理 (ZF是最方便的基本公理集之一。 然而，自从Kurt Goedel在1989年的工作以来， 20世纪30年代，没有公理集理论可以完成，以及 一致 这意味着没有一套公理可以是强大的 足以证明每一个可能的命题或其否定， 但不是两者都有。 在这个惊人的定理上， 理论，处理可能的命题P，使得P或Not P可以添加到ZF中而不会产生矛盾。 任何 这样的命题P被认为是与ZF无关的，并且可以采用 作为集合论的附加公理。 主要技术为 找到这样的独立命题是保罗·科恩所谓的 强迫的方法及其后代。 这就是思想的循环 参与并激励研究者的研究。

Friedman intends to investigate four areas within set theory in addition to beginning work on a book on the method of L-coding. (The latter should help make the many significant applications of this method accessible to a wider audience of set-theorists.) (1) With his former student, Dorshka Wylie (now an NSF fellow and an instructor at M.I.T.), he is engaged in a radically new approach to the core model K for a strong cardinal, which should provide a powerful new tool for establishing combinatorial principles in K. (2) A second goal is to enable coding methods to lift smoothly to K, permitting extension of results such as his proof of the Pi-1-2- Singleton Conjecture to the core model context. (3) A long- standing open question in the theory of iterated forcing concerns the problem of extending the method of countable support iteration beyond omega-2 without collapsing cardinals. A number of important consistency problems could be solved by such a technique. With his student, Dimitiros Tzimas, Friedman will attempt to develop such a method through the use of morasses. (4) Friedman will also explore questions in descriptive set theory as to whether the full structure theory for projective sets can be developed directly from the existence of large cardinals, without the use of infinite game theory. Set theory provides the popular way to lay the foundations for all of mathematics, the best known systematic attempt to do this being Russell and Whitehead's Principia Mathematica, dating from the early part of the 20th Century. Important foundational questions concern the independence and the consistency of the axioms used to establish set theory. Zermelo and Fraenkel's axioms (ZF for short) are one of the most convenient sets of basic axioms. However, it has been known since the work of Kurt Goedel in the 1930's that no axioms for set theory can be complete as well as consistent. This means that no set of axioms can be powerful enough to prove every possible proposition or else its negation, but not both. Upon this startling theorem has been erected a rich theory, treating possible propositions P such that either P or Not P can be added to ZF without resulting in a contradiction. Any such proposition P is said to be independent of ZF and can be taken as an additional axiom of set theory. The principal technique for finding such independent propositions is Paul J. Cohen's so-called method of forcing and its offspring. This is the circle of ideas involved in and motivating the investigator's research.