Mathemtical Sciences: RUI: Problems in Forcing

9505157 Stanley这个项目包括继续研究类强迫和非泛型实数，一些关于应用于唯一泛型对象的固定集的组合问题，以及强迫中的其他问题。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》。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一致又完整。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。任何这样的命题都被认为是独立于ZF的，并且可以作为集合论的附加公理。寻找这类独立公理的主要方法是保罗·j·科恩（Paul J. Cohen）所谓的强迫法及其衍生法。这是一个涉及和激励研究者进行研究的思想圈。＊＊＊

9505157 Stanley The project involves continued work on class forcing and non-generic reals, some combinatorial questions regarding stationary sets that have application to unique generic objects, and other problems in forcing. 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. 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 set 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 consistent as well as complete. 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 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 axioms 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. ***