Iteration Trees

large cardinal hypothesis used to prove I1~+1 determinacy in [MS] cannot be substantially improved. Our theorem is an application of the theory of core models or canonical inner models for large cardinal hypotheses. Our work develops further some of the basic ideas of this theory. The theory itself has many other applications, particularly in the realm of relative consistency results. (As a quotable paradigm for a broad class of results, let us mention the following theorem of Solovay and Shelah: ZF + DC + "All sets of reals are Lebesgue measurable" is consistent if and only if ZFC + "There is an inaccessible cardinal" is consistent. In the "only if' direction, the model that one produces of ZFC + "There is an inaccessible cardinal" is a core model and, in fact, the smallest core model, Godel's L.) We shall begin with a general description of core model theory, of some of its history, and of the advance we have made. Many statements of mathematical interest cannot be decided in ZFC, the commonly accepted system of axioms for set theory. There are many widely varying and mutually incompatible ways of removing parts of this incompleteness. Among the multitude of corresponding models of ZFC, however, the class of core models stands out. The models in this class have a high degree of resemblance to one another, and there are powerful methods which produce a detailed account of their structure. On the other hand, the class seems rich enough to