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Harrington's principle over higher order...
Cheng, Yong...

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Harrington's principle over higher order arithmetic by Cheng, Yong ( Author )
Lemonjar Open Access
N.A
13-03-2015
Let Z2, Z3, and Z4 denote 2nd, 3rd, and 4th 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(Σ11) implies 0♯ exists" are done in two steps: first show that Det(Σ11) implies {\sf HP}, and then show that {\sf 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.
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MYR 0.01
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https://arxiv.org/abs/1503.04000
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