Mathematical Sciences: Forcing and O#

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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.