Harrington's principle over higher order arithmetic

Abstract

Let $Z_2$, $Z_3$, and $Z_4$ denote $2^{\rm nd}$, $3^{\rm rd}$, 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 "$Det(\Sigma_1^1)$ implies $0^{\sharp}$ exists" are done in two steps: first show that $Det(\Sigma_1^1)$ implies {\sf HP}, and then show that {\sf HP} implies $0^{\sharp}$ exists. The first step is provable in $Z_2$. In this paper we show that $Z_2 \, + \, {\sf HP}$ is equiconsistent with ${\sf ZFC}$ and that $Z_3\, + \, {\sf HP}$ is equiconsistent with ${\sf ZFC} \, +$ there exists a remarkable cardinal. As a corollary, $Z_3\, + \, {\sf HP}$ does not imply $0^{\sharp}$ exists, whereas $Z_4\, + \, {\sf HP}$ does. We also study strengthenings of Harrington's Principle over $2^{\rm nd}$ and $3^{\rm rd}$ order arithmetic.