Strengthenings of Harrington’s Principle

  • Yong ChengEmail author
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)


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.


  1. 1.
    Cheng, Y., Schindler, R.: Harrington’s Principle in higher-order arithmetic. J. Symb. Log. 80(02), 477–489 (2015)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Harrington, L.A., Baumgartner, J.E., Kleinberg, E.M.: Adding a closed unbounded set. J. Symb. Log. 41, 481–482 (1976)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Jensen, R.B.: Lecture note on subcomplete forcing and L-forcing. Handwritten notes.
  4. 4.
    Shelah, S.: Proper and Improper Forcing. Perspectives in Math. Logic. Springer (1998)Google Scholar
  5. 5.
    Cummings, J.: Iterated forcing and elementary embeddings. In: Foreman, M., Kanamori, A. (Eds.) Chapter 12 in Handbook of Set Theory. Springer, Berlin (2010)Google Scholar
  6. 6.
    Jensen, R.B.: Iteration Theorems for Subcomplete and Related Forcings. Handwritten notes.

Copyright information

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

Authors and Affiliations

  1. 1.School of PhilosophyWuhan UniversityWuhanChina

Personalised recommendations