ON THE FIRST n STRONGLY COMPACT CARDINALS

Using techniques of Kimchi and Magidor, we generalize an earlier result and show that it is relatively consistent for the first n strongly compact cardinals to be somewhat supercompact yet not fully supercompact. The class of strongly compact cardinals is, without a doubt, one of the most peculiar in the entire theory of large cardinals. As is well known, the class of strongly compact cardinals suffers from a severe identity crisis. Magidor's fundamental result of [Ma] shows that it is consistent for the least strongly compact cardinal to coincide with either the least measurable or the least supercompact cardinal. The result of [Al] shows that it is consistent for the least strongly compact cardinal to be somewhat supercompact although not fully supercompact. The result of Kimchi and Magidor [KM] shows that it is consistent for the class of strongly compact cardinals to coincide with the class of supercompact cardinals (except at limit points where Menas' result [Me] shows that such a coincidence may not occur) or for the first n for n £ œ strongly compact cardinals to coincide with the first n measurable cardinals. The purpose of this paper is to show that matters can be muddled still further. Specifically, we generalize the result of [Al] using the methods of [KM] and prove the following Theorem. Let V \= "K = {kx , ... , k„} with kx < k2 < ■■■ < k„ the first n supercompact cardinals". For each k g K, let K, Vt "0 = Ki(S) > <pKj(S). There is then a partial ordering P so that for each k £ K, Vp 1= " k is Received by the editors March 14, 1991 and, in revised form, November 29, 1993. 1991 Mathematics Subject Classification. Primary 03E35, 03E55. The research for this paper was partially supported by PSC-CUNY Grants 661371 and 662341 and by a salary grant from Tel Aviv University. ©1995 American Mathematical Society