Forcing a set model of Z3 + Harrington's Principle

Let Z3 denote third order arithmetic. Let Harrington's Principle, HP , denote the statement that there is a real x such that every x‐admissible ordinal is a cardinal in L . In this paper, assuming there exists a remarkable cardinal with a weakly inaccessible cardinal above it, we force a set model of Z3+HP via set forcing without reshaping.