Strengthenings of Harrington’s Principle

Abstract

In this chapter, we examine the large cardinal strength of strengthenings of Harrington’s Principle, $\mathsf{HP(\varphi )}$ , over $\mathsf{Z_2}$ and $\mathsf{Z_3}$ . In Sect. 4.3, we prove that $\mathsf{Z_2} + \mathsf{HP}(\varphi )$ is equiconsistent with “$\mathsf{ZFC} + \{\alpha | \varphi (\alpha )\}$ is stationary”. In Sect. 4.5, we prove that $\mathsf{Z_3} + \mathsf{HP}(\varphi )$ is equiconsistent with “$\mathsf{ZFC} +$ there exists a remarkable cardinal $\kappa$ with $\varphi (\kappa ) + \{\alpha | \varphi (\alpha )\wedge \{\beta <\alpha | \varphi (\beta )\}$ is stationary in $\alpha$} is stationary”. As a corollary, $\mathsf{Z_4}$ is the minimal system of higher-order arithmetic for proving that $\mathsf{HP}, \mathsf{HP}(\varphi )$, and $0^{\sharp }$ exists are pairwise equivalent with each other.