Partial near supercompactness
The author introduces a refinement of the levels of supercompactness, called near supercompactness. A cardinal \(\kappa\) is \textit{nearly \(\lambda\)-supercompact} if for every \(A \subset \lambda\) there is a transitive model \(M\) of ZFC minus the power set axiom (with collection used instead of replacement), closed under sequences of length less than \(\kappa\) and containing \(A\), \(\kappa\), and \(\lambda\), and a transitive \(N\), and an elementary embedding \(j: M \to N\) with critical point \(\kappa\) such that \(j(\kappa) > \lambda\) and \(j''\lambda \in N\). It interleaves as follows: any cardinal that is \(\lambda\)-supercompact is also nearly \(\lambda\)-supercompact, and conversely a cardinal that is nearly \(2^{\lambda^{<\kappa}}\)-supercompact is \(\lambda\)-supercompact. The author then generalizes indestructibility results as well as the construction of a Laver diamond to this new hierarchy.
scientific article; zbMATH DE number 6111171