HARRINGTON'S PRINCIPLE IN HIGHER ORDER ARITHMETIC

Let Z(2), Z(3), and Z(4) 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(Sigma(1)(1)) implies 0(#) exists are done in two steps: first show that Det(Sigma(1)(1)) implies HP, and then show that HP implies 0(#) exists. The first step is provable in Z(2). In this paper we show that Z(2) + HP is equiconsistent with ZFC and that Z(3) + HP is equiconsistent with ZFC + there exists a remarkable cardinal. As a corollary, Z(3) + HP does not imply Ol exists, whereas Z(4) +Y HP does. We also study strengthenings of Harrington's Principle over 2nd and iird order arithmetic.