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Strengthenings of Harrington’s Principle

  • Yong ChengEmail author
Chapter
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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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Copyright information

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.School of PhilosophyWuhan UniversityWuhanChina

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