Harrington's principle over higher order arithmetic

Abstract

Let $Z_2$, $Z_3$, and $Z_4$ denote $2^{\mathrm{nd}}$, $3^{\mathrm{rd}}$, and $4^{\mathrm{th}}$ 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 "$Det({\Sigma }_1^1)$ implies $0^{♯}$ exists" are done in two steps: first show that $Det({\Sigma }_1^1)$ implies {\sf HP}, and then show that {\sf HP} implies $0^{♯}$ exists. The first step is provable in $Z_2$. In this paper we show that $Z_2+\mathsf{HP}$ is equiconsistent with $\mathsf{ZFC}$ and that $Z_3+\mathsf{HP}$ is equiconsistent with $\mathsf{ZFC}+$ there exists a remarkable cardinal. As a corollary, $Z_3+\mathsf{HP}$ does not imply $0^{♯}$ exists, whereas $Z_4+\mathsf{HP}$ does. We also study strengthenings of Harrington's Principle over $2^{\mathrm{nd}}$ and $3^{\mathrm{rd}}$ order arithmetic.

13 pages, to appear in JSL Volume 80,Issue 2 (June 2015)

Cited by in corpus (4)

• Current research on Gödel's incompleteness theorems
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• The strong reflecting property and Harrington's Principle
• Force a set model of $Z_3$ + Harrington's Principle