“ On the Strong Equality between Supercompactness and Strong Compactness ” by Arthur

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. *The research of the first author was partially supported by PSC-CUNY Grant 662341 and a salary grant from Tel Aviv University. In addition, the first author wishes to thank the Mathematics Departments of Hebrew University and Tel Aviv University for the hospitality shown him during his sabbatical in Israel. **Publication 495. The second author wishes to thank the Basic Research Fund of the Israeli Academy of Sciences for partially supporting this research.