Mathemtical Sciences: RUI: Problems in Forcing

9505157斯坦利这个项目涉及到关于类强迫和非类属实数的持续工作，一些关于适用于独特一般对象的静止集的组合问题，以及强迫中的其他问题。集合论提供了为所有数学奠定基础的流行方法，最著名的系统尝试是罗素和怀特黑德的数学原理。重要的基础性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公理(简称ZF)是最方便的一组基本公理。然而，自从哥德尔在1930年S的工作以来，人们就知道，集合论的任何公理都不可能既一致又完整。这意味着，没有一组公理能够强大到足以证明每一个可能的命题或其否定，但不能两者兼而有之。在这个惊人的定理的基础上，建立了一个丰富的理论，处理可能的命题P，使P或不P可以加到ZF上，而不会导致矛盾。任何这样的命题都可以说是独立于ZF的，可以作为集合论的一个附加公理。寻找这种独立公理的主要技术是保罗·J·科恩的所谓强迫方法及其产物。这就是参与和激励调查者研究的思想圈子。***

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. ***