A Note on Strong Compactness and Supercompactness

In this note, we provide a new proof of Magidor's theorem that the least strongly compact cardinal can be the least supercompact cardinal. A classical theorem of Magidor [5] states that it is consistent, relative to the existence of a supercompact cardinal, for the least supercompact cardinal to be the least strongly compact cardinal. There are currently two different proofs of this fact—the original proof of [5] and the proof of Kimchi and Magidor [4], using nonreflecting stationary sets, which shows that the class of strongly compact cardinals can coincide with the class of supercompact cardinals except at measurable limits of supercompact (strongly compact) cardinals. (It is a theorem of Menas [6] that a measurable limit of strongly compact cardinals need not be supercompact.) In this note, we give a third proof of this fact which makes use of Radin forcing. To start the proof, let Kh='/c is a supercompact cardinal'. Without loss of generality, we can assume that V\= 2 ^ /c'; if this is not initially true, then we can do a reverse Easton forcing (see [1] or [3]) to make it true. Let now j : V-+ M be a supercompact elementary embedding with critical point K so that for X = \p(p(K))\,M ^ M. Using j , as in [7] we define a Radin sequence fi of measures as follows: