Harrington's principle over higher order arithmetic

Let $Z_2$, $Z_3$, and $Z_4$ denote $2^{\mathrm{n}\mathrm{d}}$, $3^{\mathrm{r}\mathrm{d}}$, and $4^{\rm th}$ order arithmetic, respectively. We let Harrington’s Principle, {\sf HP}, denote the statement that there is a real $x$ such that every $x$–admissible ordinal is a cardinal in $L$. The known proofs of Harrington’s theorem “$Det({\mathrm{\Sigma }}_1^1)$ implies $0^{\mathrm{♯}}$ exists” are done in two steps: first show that $Det({\mathrm{\Sigma }}_1^1)$ implies {\sf HP}, and then show that {\sf HP} implies $0^{\mathrm{♯}}$ exists. The first step is provable in $Z_2$. In this paper we show that $Z_2+\mathsf{H}\mathsf{P}$ is equiconsistent with $\mathsf{Z}\mathsf{F}\mathsf{C}$ and that $Z_3+\mathsf{H}\mathsf{P}$ is equiconsistent with $\mathsf{Z}\mathsf{F}\mathsf{C}+$ there exists a remarkable cardinal. As a corollary, $Z_3+\mathsf{H}\mathsf{P}$ does not imply $0^{\mathrm{♯}}$ exists, whereas $Z_4+\mathsf{H}\mathsf{P}$ does. We also study strengthenings of Harrington’s Principle over $2^{\mathrm{n}\mathrm{d}}$ and $3^{\mathrm{r}\mathrm{d}}$ order arithmetic.