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中的四个领域。（后者应该有助于使该方法的许多重要应用为集合论者的更广泛受众所接受。）(1)他和他以前的学生Dorshka Wylie（现在是NSF的研究员和麻省理工学院的讲师）正在研究一种全新的方法来研究强基数的核心模型K，这将为在K中建立组合原理提供一个强大的新工具。(2)第二个目标是使编码方法能够顺利地提升到K。允许将结果（例如他对Pi-1-2-单例猜想的证明）扩展到核心模型上下文。(3)迭代强迫理论中一个长期存在的开放性问题是将可数支持迭代方法扩展到-2以外而不坍缩基数的问题。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里罗斯·齐马斯（Dimitiros Tzimas）一起，尝试通过使用泥沼来发展这样一种方法。(4)弗里德曼还将探讨描述性集合论中的问题，即投影集的完整结构理论是否可以直接从大基数的存在中发展出来，而不使用无限博弈论。集合论为所有数学奠定了基础，最著名的系统尝试是罗素和怀特黑德的《数学原理》，可以追溯到20世纪早期。重要的基础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的公理（简称ZF）是最方便的基本公理集之一。然而，自从20世纪30年代库尔特·哥德尔（Kurt Goedel）的工作以来，人们已经知道集合论的公理不可能既完整又一致。这意味着没有一组公理能强大到足以证明每一个可能的命题或否定，但不能两者都证明。在这个惊人的定理的基础上，建立了一个丰富的理论，处理可能命题P，使得P或非P都可以加到ZF中而不会产生矛盾。任何这样的命题P都是独立于ZF的，并且可以作为集合论的附加公理。寻找这些独立命题的主要技术是保罗·j·科恩所谓的强迫方法及其衍生方法。这是一个涉及和激励研究者进行研究的思想圈。

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.