Mathematical Sciences: Forcing and O#

研究者打算继续研究由0构造的泛型对象和其他强迫问题。特别地，他将尝试推广S. D. Friedman的工作，得到一个非可构造的zf - pi -1-2单态可构造。集合论中的几个相关问题也将被考虑。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完整又一致。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P都是独立于ZF的，并且可以作为集合论的附加公理。寻找这类独立命题的主要技术是保罗·j·科恩的所谓强迫法及其衍生法。这是一个涉及和激励研究者进行研究的思想圈。

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.