The Minimal System of Higher Order Arithmetic to Show That Harrington’s Principle Implies 0 Exists
نویسندگان
چکیده
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. Harrington’s proof of “Det(Σ1) implies 0 ] exists” in ZF is done in two steps: first show that Det(Σ1) implies HP and secondly show that HP implies 0 ] exists. We observe that the first step is provable in Z2. In this paper we prove via class forcing that Z2 + HP is equiconsistent with ZFC and Z3 + HP is equiconsistent with ZFC+ there exists a remarkable cardinal. As a corollary, we have Z3 + HP does not imply 0] exists. We also observe that Z4 + HP implies 0] exists. As a corollary, Z4 is the minimal system of higher order arithmetic to show that HP implies 0] exists.
منابع مشابه
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 ...
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