CAREER: Large Cardinals

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The PI is investigating several aspects of large cardinal theory in theregion of Woodin cardinals, including (1) Long Games; (2) Iterability; and(3) connections with Descriptive Set Theory. 1 - Results obtained by thePI through previous NSF support demonstrate the tight connection betweenWoodin cardinals and games of variable countable length. The existence ofiterable models for specific large cardinals can be used to yield thedeterminacy of games of specific lengths, with increasing lengthcorresponding to increasing large cardinal strength. These games in turncan be used to decide statements over minimal inner models for largecardinals. The PI is attempting to extend this correspondence, the firstmain test case being open games of length omega one. 2 - The study ofWoodin cardinals leads naturally to a transfinite game known as theiteration game. Winning strategies ("iteration strategies") for the goodplayer in this game are essential to the comparison process that lies atthe heart of inner model theory. Proving that iteration strategies existis the single most important problem in the field. The PI has been workingon this problem, contributing to the existing pool of partial results.Several problems, related to the general problem of iterability butspecialized and more concrete, are investigated as part of this project.This investigation should provide increased understanding of the mainproblem, and hopefully lead to additional partial results. 3 - The PI isworking to solidify the connections between large cardinals and definablesets of real numbers. Extensive work, much of it done during the '70s and'80s, provides a detailed analysis of definable sets of reals assumingdeterminacy. Later work obtained determinacy from large cardinals. The PIis working to emulate the existing analysis of definable sets of reals,working directly from large cardinals. The point here is to try to convertmethods used in the study of definable sets of reals, into methods whichcan be used in inner model theory and the study of large cardinals.Set Theory is a branch of Mathematics which attempts to understand theuniverse of Mathematics, that is the collection of all objects studied byMathematicians, or more precisely the collection of all _sets_. (Numbers,groups, functions, etc., can all be represented as sets.) Set Theoristsview this universe as a structure in its own right, a structure which canitself be analyzed using mathematical reasoning. For example, a SetTheorist may ask "is there an embedding of the universe into a similar,yet not identical, structure?" Set Theorists work with such embeddings("elementary embeddings") in much the same way that one would work withfunctions on numbers, asking "how similar is the target structure to theoriginal structure?" and "what's the smallest size of a set actually movedby the embedding?" Such questions about embeddings of the universe form asubfield of Set Theory known as the study of large cardinals. Theterminology here hints at the answer to the last question mentioned.Indeed, the sets actually moved by elementary embeddings are substantiallylarger than any object studied in other branches of Mathematics(substantially larger than the set of real numbers for example). Yet itturns out that the existence of elementary embeddings has concrete effectson lower level objects, including concrete effects on real numbers. Thisproject is part of the on-going effort to better understand largecardinals, and better understand their effects on the line of realnumbers. The PI, like other researchers in the field, is motivated by theconnection between the abstract and the concrete.