Mathematical Sciences: Set Theory

除了开始写一本关于L编码方法的书外，弗里德曼还打算研究集合论的四个领域。(后者应该有助于使这种方法的许多重要应用为更广泛的集合论者所接受。)(1)与他以前的学生Dorshka Wylie(现在是NSF研究员和麻省理工学院的讲师)一起，他对核心模型K进行了一种全新的方法，作为一名强大的红衣主教，(2)第二个目标是使编码方法能够平滑地提升到K，从而允许将结果(例如他对PI-1-2-Singleton猜想的证明)扩展到核心模型上下文。(3)迭代强迫理论中一个长期悬而未决的问题是在不折叠基数的情况下将可数支承迭代方法扩展到omega-2以外的问题。这种技术可以解决许多重要的一致性问题。弗里德曼将与他的学生迪米特罗斯·齐马斯一起，试图通过使用沼泽来开发这种方法。(4)Friedman还将探讨描述性集合论中的问题，即射影集的全结构理论是否可以直接从大基数的存在发展而不使用无限博弈论。集合论提供了为所有数学奠定基础的流行方法，最著名的系统尝试是罗素和怀特海的《数学原理》，可以追溯到20世纪初。重要的基础性问题涉及用于建立集合论的公理的独立性和一致性。Zermelo和Fraenkel公理(简称ZF)是最方便的基本公理之一。然而，自从哥德尔在20世纪30年代的S的工作以来，人们就知道，集合论的任何公理都不可能既完整又一致。这意味着，没有一组公理能够强大到足以证明每一个可能的命题或其否定，但不能两者兼而有之。在这个惊人的定理的基础上，建立了一个丰富的理论，处理可能的命题P，使P或不P可以加到ZF上，而不会导致矛盾。任何这样的命题P都可以说是独立于ZF的，并且可以作为集合论的一个附加公理。寻找这种独立命题的主要技术是保罗·J·科恩的所谓强迫方法及其产物。这就是参与和激励调查者研究的思想圈子。

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.