Strong compactness, measurability, and the class of supercompact cardinals (Q2717642)

In this paper, the author continues the investigation of the possible interplays of supercompactness, strong compactness and measurability. The author shows how to achieve simultaneously the following three properties when designing forcing extensions: (1) preserve the supercompactness of all those supercompact cardinals which are limits of supercompact cardinals; (2) kill the weak compactness of all those supercompact cardinals which are not limits of supercompact cardinals; (3) preserve enough strong compactness so that each surviving supercompact cardinal is a limit of strongly compact cardinals. The author also shows how to preserve supercompactness and at the same time make measurability equal to certain strong compactness.