A new Easton theorem for supercompactness and level by level equivalence
We establish a new Easton theorem for the least supercompact cardinal $\kappa $ that is consistent with the level by level equivalence between strong compactness and supercompactness. This theorem is true in any model of ZFC containing at least one supercompact cardinal, regardless if level by level equivalence holds. Unlike previous Easton theorems for supercompactness, there are no limits on the Easton functions $F$ used, other than the usual constraints given by Easton’s theorem and the fact that if $\delta \lt \kappa $ is regular, then $F(\delta ) \lt \kappa $. In both our ground model and the model witnessing the conclusions of our theorem, there are no restrictions on the structure of the class of supercompact cardinals.