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Abstract

We force and construct a model in which level by level equivalence between strong compactness and supercompactness holds, along with certain additional “inner model like” properties. In particular, in this model, the class of Mahlo cardinals reflecting stationary sets is the same as the class of weakly compact cardinals, and every regular Jónsson cardinal is weakly compact. On the other hand, we force and construct a model for the level by level equivalence between strong compactness and supercompactness in which on a stationary subset of the least supercompact cardinal $\kappa $, there are non-weakly compact Mahlo cardinals which reflect stationary sets. We also examine some extensions and limitations on what is possible in our theorems. Finally, we indicate how to ensure in our models that $\diamondsuit _\delta $ holds for every successor and Mahlo cardinal $\delta $, and below the least supercompact cardinal $\kappa $, $\square _\delta $ holds on a stationary subset of $\kappa $. There are no restrictions in our main models on the structure of the class of supercompact cardinals.