Mathematical Sciences: Set Theory

DMS 9625997 Sy D.Friedman Friedman打算在四个方向上工作，所有这些工作都旨在通过精细结构和强迫技术来理解集合论宇宙的结构。首先，他打算完成他的书，名为《精细结构和阶级强迫》。其次，弗里德曼将通过使用编码方法和他的迭代类强迫方法对L在类强迫领域中的一些问题进行追问。第三，Friedman打算为一个与K=强基数的核心模型相关的类似理论奠定基础，首先为这个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共同开展。最后，弗里德曼将研究其他一些强迫问题，与0#有关(与Velickovic有关)，涉及大基数(与Cummings有关)和可定义性理论(与Bagaria有关)。本世纪初，数学家发展了集合论公理系统ZFC，足以表达和证明数学定理。然而，ZFC是不完整的：这一理论中有一些既不能证明也不能反驳的说法。对于集合论者来说，一项重要的任务就是发现新的公理来补充ZFC，以减轻这种不完全性。这将为工作中的数学家提供强大的新工具，这可能还会对一般的科学具有令人兴奋的应用。弗里德曼的工作集中在对ZFC模型结构的尽可能深入的理解上，作为发现正确的公理来补充这一理论的一种方式。

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.