Mathematical Sciences: Set Theory

弗里德曼打算在四个方向上工作，所有的目标都是通过精细结构和强迫的技术来理解集合论宇宙的结构。首先，他打算完成他的著作《精细结构与阶级强迫》。其次，Friedman将通过使用编码方法和他的迭代类强制方法，在L的类强制领域中寻求一些问题。第三，Friedman打算为一个相对于K=强基数的核心模型的类似理论奠定基础，首先为该模型发展一个改进的精细结构理论，然后使编码方法适应这个新的背景。这项工作将与他的合作者Jensen， Koepke和Wylie共同进行。最后，弗里德曼将研究一些其他强迫问题，与0有关（与维利科维奇），涉及大基数（与卡明斯）和可定义性理论（与巴格里亚）。本世纪初，数学家发展了集合论公理体系ZFC，足以表达和证明数学定理。然而，ZFC是不完整的：在这个理论中有一些既不能证明也不能反驳的陈述。对于集合理论家来说，一个重要的任务是发现新的公理来与ZFC相邻，以减轻这种不完备性。这将为工作的数学家提供强大的新工具，此外，它可能在一般科学中有令人兴奋的应用。弗里德曼的工作重点是尽可能深入地理解ZFC模型的结构，作为发现正确公理的一种方式来添加到这个理论中。

DMS 9625997 Sy D. Friedman Friedman intends to work in four directions, all aimed at an understanding of the structure of the set-theoretic universe through the techniques of fine structure and forcing. First, he intends to complete his book, entitled, "Fine Structure and Class Forcing". Second, Friedman will pursue a number of questions in the area of class forcing over L through the use of the coding method and his method of iterated class forcing. Third, Friedman intends to lay the foundation for an analogous theory relative to K=the core model for a strong cardinal, by first developing an improved fine structure theory for this model, and then adapting the coding method to this new context. This work will be carried out jointly with his collaborators Jensen, Koepke and Wylie. Lastly, Friedman will investigate a number of other forcing problems, related to 0# (with Velickovic), involving large cardinals (with Cummings) and in definability theory (with Bagaria). Early this century, mathematicians developed the system ZFC of axioms for set theory, adequate to express and prove the theorems of mathematics. However ZFC is incomplete: there are statements that can neither be proved nor refuted in this theory. An important task for the set theorist is to uncover new axioms to adjoin to ZFC so as to alleviate this incompleteness. This will provide powerful new tools for the working mathematician, which may in addition have exciting applications for science in general. Friedman's work is focused on the deepest possible understanding of the structure of models of ZFC, as a way of discovering the right axioms to add to this theory.