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. ***
