Journal
JOURNAL OF SYMBOLIC LOGIC
Volume 80, Issue 2, Pages 477-489Publisher
CAMBRIDGE UNIV PRESS
DOI: 10.1017/jsl.2014.31
Keywords
Harrington's Principle HP; 0(#); remarkable cardinal; almost disjoint forcing; class forcing; reshaping; HP(phi); subcomplete forcing; Revised Countable Support (RCS) iterations; iterated club shooting; Z(2), Z(3), Z(4)
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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.
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