Consistency Results concerning Supercompactness

A general framework for proving relative consistency results with regard to supercompactness is developed. Within this framework we prove the relative consistency of the assertion that every set is ordinal definable with the statement asserting the existence of a supercompact cardinal. We also generalize Easton's theorem; the new element in our result is that our forcing conditions preserve supercompactness. Introduction. The framework for our results is "backward Easton forcing": forcing conditions are constructed in the ground model by an iteration similar to the iteration described in the Solovay-Tennenbaum paper [12], the essential difference being that at the limit stages of the construction one takes the inverse limit (instead of the direct limit) of the conditions constructed at the previous stages. Backward Easton forcing is independent from large cardinal theory. Indeed, large cardinals are mentioned only in the latter part of this paper. The concept of supercompactness is due to Solovay [7]. We shall need only the most elementary facts concerning supercompact cardinals, which we provide in §0. The essential idea of the backward Easton forcing constructions is probably due to R. Jensen [unpublished, 1965]. A few years later and independently of Jensen's work, F. Tall used similar constructions to obtain the consistency of various conjectures in topology [14]. J. Silver realized the importance of these methods to the theory of large cardinals, and he refined and extended them to a method, to which we refer as the "Silver forcing method", for preserving certain large cardinal properties in suitable Cohen extensions. By this method Silver Received by the editors September 9, 1974. AMS (MOS) subject classifications (1970). Primary 02K05, 02K35.