Mathematical Sciences: Forcing and O#

研究人员打算继续研究由0构造的一般对象和强迫中的其他问题。特别是，他将尝试扩展S.D.Friedman的工作，得到一个从0开始不可构造的ZF-PI-1-2-Singleton。还将考虑集合论中的几个相关问题。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)是最方便的基本公理之一。然而，自从哥德尔在20世纪30年代的S的工作以来，人们就知道，集合论的任何公理都不可能既完整又一致。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P都可以说是独立于ZF的，并且可以作为集合论的一个附加公理。寻找这种独立命题的主要技术是保罗·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。

The investigator intends to continue work on generic objects constructed from 0 and other problems in forcing. In particular, he will attempt to extend S. D. Friedman's work, obtaining a non-constructible ZF-Pi-1-2-singleton constructible from 0. Several related problems in set theory will also be considered. 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 sets 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 complete as well as consistent. 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 P 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 propositions 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.