On the Strong Equality between Supercompactness and Strong Compactness

We show that supercompactness and strong compactness can be equivalent even as properties of pairs of regular cardinals. Specifically, we show that if V |= ZFC + GCH is a given model (which in interesting cases contains instances of supercompactness), then there is some cardinal and cofinality preserving generic extension V [G] |= ZFC + GCH in which, (a) (preservation) for κ ≤ λ regular, if V |= “κ is λ supercompact”, then V [G] |= “κ is λ supercompact” and so that, (b) (equivalence) for κ ≤ λ regular, V [G] |= “κ is λ strongly compact” iff V [G] |= “κ is λ supercompact”, except possibly if κ is a measurable limit of cardinals which are λ supercompact. 0. Introduction and Preliminaries It is a well known fact that the notion of strongly compact cardinal represents a singularity in the hierarchy of large cardinals. The work of Magidor [Ma1] shows that the least strongly compact cardinal and the least supercompact cardinal can coincide, but also, the least strongly compact cardinal and the least measurable cardinal can coincide. The work of Kimchi and Magidor [KiM] generalizes this, showing that the class of strongly compact cardinals and the class of supercompact cardinals can coincide (except by results of Menas [Me] and [A] at certain measurable limits of supercompact cardinals), and the first n strongly compact cardinals (for n a natural number) and the first n measurable cardinals can coincide. Thus, the precise identity of certain members of the class of strongly compact cardinals cannot be ascertained vis à vis the class of measurable cardinals or the class of supercompact cardinals. An interesting aspect of the proofs of both [Ma1] and [KiM] is that in each result, all “bad” instances of strong compactness are not obliterated. Specifically, in each model, since the strategy employed in destroying strongly compact cardinals which aren’t also supercompact is to make them non-strongly compact after a certain point either by adding a Prikry sequence or a non-reflecting stationary set of ordinals of the appropriate cofinality, there may be cardinals κ and λ so that κ is λ strongly Received by the editors May 2, 1994 and, in revised form, December 30, 1994. 1991 Mathematics Subject Classification. Primary 03E35; Secondary 03E55.