Harrington's Principle in Higher order Arithmetic

Let Z2, Z3, and Z4 denote 2nd, 3rd, and 4th order arithmetic, respectively. We 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. The known proofs of Harrington’s theorem “Det(Σ1) implies 0 ] exists” are done in two steps: first show that Det(Σ1) implies HP, and then show that HP implies 0] exists. The first step is provable in Z2. In this paper we show that Z2 + HP is equiconsistent with ZFC and that Z3 + HP is equiconsistent with ZFC+ there exists a remarkable cardinal. As a corollary, Z3 + HP does not imply 0] exists, whereas Z4 + HP does. We also study strengthenings of Harrington’s Principle over 2nd and 3rd order arithmetic.