Mathematical Sciences: Set Theory

数学科学：集合论

• 批准号: 9625997
• 负责人: Sy Friedman
• 金额: --
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-08-01 至 1999-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9625997&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Set; Theory

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.

DMS 9625997 Sy D.弗里德曼 弗里德曼打算在四个方向上工作，所有的目的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，他打算完成他的书，题为“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，在L上的类强迫领域中提出一些问题。 第三，弗里德曼打算为一个与K=强基数的核心模型类似的理论奠定基础，首先为这个模型开发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詹森，科普克和怀利共同进行。 最后，弗里德曼将研究一些其他的强迫问题，与0#有关（与Velickovic），涉及大基数（与Cummings）和可定义性理论（与Bagaria）。 早在这个世纪，数学家发展了系统ZFC公理集理论，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，有些陈述既不能被证明也不能被反驳。 集合论的一个重要任务是发现新的公理来邻接ZFC，以减轻这种不完备性。 这将为数学工作者提供强大的新工具，这也可能对一般科学有令人兴奋的应用。 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。