# Strengthenings of Harrington’s Principle

• Yong Cheng
Chapter
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)

## 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.

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