Force a set model of $Z_3$ + Harrington's Principle

Abstract

Let $Z_3$ denote $3^{rd}$ 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 $Z_3\, + \, {\sf HP}$ via set forcing without reshaping.