Some Applications of ~terated Ultrapowers in Set Theory

Let 91 be a ~-complete ultrafilter on the measurable cardinal ~. Scott [ 1 3 ] proved V ¢ L by using 91 to take the ultrapower of V. Gaifman [ 2] considered iterated ultrapowers of V by cg to conclude even stronger results; for example, that L n ~(6o) is count-able. In this paper we discuss some new applications of iterated ultra-powers. In § § 1-4, we develop a straightforward generalization of Gaif-man's method which is needed fo~ some of the restqts in § § 6-1 1. Namely, we consider iterated uhrapowers of a sub-model, M, of the universe by an ultrafilter which need not be in M. § 5 discusses some known results within our present framework. In § 6, we investigate the universe constructed from a normal ultrafilter on the measurable cardinal ~:, and show that in this universe tb~ normal ultrafilter is unique. In § 7, we obtain a character-~zation of arbitrary ~-complete free ultrafilters in this universe, and in § 8, we show that this universe has some pathological model-theoretic properties. § 9 uses methods of § 6 to discuss the problem of GCH at a measurable cardinal. We show that in the theory