Forcing level by level equivalence and a consequence of UA

We show how Hamkins’ Gap Forcing Theorem of Hamkins (Israel J Math 125:237–252, 2001, Bull Symb Logic 5: 264–272, 1999) can be used to give an alternate construction of models for the level by level equivalence between strong compactness and supercompactness when forcing over models of ZFC containing one supercompact cardinal in which no cardinal is supercompact up to a measurable cardinal. As an application of our methods, we also show that starting from such a model V which in addition satisfies Goldberg’s Ultrapower Axiom UA, it is possible to force and construct a model M with a supercompact cardinal \(\kappa \) $\kappa$ satisfying level by level equivalence having certain additional properties. In particular, in M, no cardinal is supercompact up to a measurable cardinal. there is a stationary subset of measurable cardinals \(A \subseteq \kappa \) $A\subseteq \kappa$ such that for every \(\delta \in A\) $\delta \in A$ , \((o(\delta ))^V < \delta ^{++}\) ${(o(\delta ))}^V<{\delta }^{++}$ , \((o(\delta ))^V = (o(\delta ))^M\) ${(o(\delta ))}^V={(o(\delta ))}^M$ , and the Mitchell ordering of normal measures over \(\delta \) $\delta$ is linear. M can contain inaccessible and Mahlo cardinals above \(\kappa \) $\kappa$ . It is currently unknown whether a model of ZFC with a supercompact cardinal and M’s properties can also satisfy UA.