Journal of the Mathematical Society of Japan

Reflection principles for $\omega_2$ and the semi-stationary reflection principle

Toshimichi USUBA

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Starting from a model with a weakly compact cardinal, we construct a model in which the weak stationary reflection principle for $\omega_2$ holds but the Fodor-type reflection principle for $\omega_2$ fails. So the stationary reflection principle for $\omega_2$ fails in this model. We also construct a model in which the semi-stationary reflection principle holds but the Fodor-type reflection principle for $\omega_2$ fails.

Article information

Source
J. Math. Soc. Japan Volume 68, Number 3 (2016), 1081-1098.

Dates
First available in Project Euclid: 19 July 2016

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1468956160

Digital Object Identifier
doi:10.2969/jmsj/06831081

Mathematical Reviews number (MathSciNet)
MR3523539

Zentralblatt MATH identifier
06642405

Subjects
Primary: 03E35: Consistency and independence results
Secondary: 03E05: Other combinatorial set theory

Keywords
stationary reflection principle semi-stationary reflection principle Fodor-type reflection principle

Citation

USUBA, Toshimichi. Reflection principles for $\omega_2$ and the semi-stationary reflection principle. J. Math. Soc. Japan 68 (2016), no. 3, 1081--1098. doi:10.2969/jmsj/06831081. https://projecteuclid.org/euclid.jmsj/1468956160.


Export citation

References

  • J. E. Baumgartner, A new class of order types, Ann. Math. Log., 9 (1976), 187–222.
  • P. Doebler, Rado's conjecture implies that all stationary set preserving forcings are semiproper, J. Math. Log., 13 (2013), 1350001.
  • P. Doebler and R. Schindler, $\Pi_2$ consequences of $\mathrm{BMM}+\mathrm{NS}_{\omega_1}$ is precipitous and the semiproperness of stationary set preserving forcings, Math. Res. Lett., 16 (2009), 797–815.
  • M. Foreman, M. Magidor and S. Shelah, Martin's maximum, saturated ideals, and nonregular ultrafilters. I, Ann. of Math. (2), 127 (1988), 1–47.
  • S. Fuchino, I. Juhász, L. Soukup, Z. Szentmiklossy and T. Usuba, Fodor-type reflection principle and reflection of metrizability and meta-Lindelöfness, Top. Appl., 157 (2010), 1415–1429.
  • S. Fuchino and A. Rinot, Openly generated Boolean algebras and the Fodor-type Reflection Principle, Fund. Math., 212 (2011), 261–283.
  • S. Fuchino, L. Soukup, H. Sakai and T. Usuba, More about Fodor-type Reflection Principle, submitted for publication.
  • S. Fuchino, H. Sakai, V. T. Perez and T. Usuba, Rado's Conjecture and the Fodor-type Reflection Principle, in preparation.
  • B. Koenig, P. Larson and Y. Yoshinobu, Guessing clubs in the generalized club filter, Fund. Math., 195 (2007), 177–191.
  • J. Krueger, On the weak reflection principle, Trans. Amer. Math. Soc., 363 (2011), 5537–5576.
  • T. Miyamoto, On the consistency strength of the FRP for the second uncountable cardinal, RIMS Kôkyûroku, 1686 (2010), 80–92.
  • H. Sakai, Partial stationary reflection in ${\mathcal P}_{\omega_1} \omega_2$, RIMS Kôkyûroku, 1595 (2008), 47–62.
  • H. Sakai, Semistationary and stationary reflection, J. Symbolic Logic, 73 (2008), 181–192.
  • S. Shelah, Proper and improper forcing. Second edition, Perspectives in Math. Log., Springer-Verlag, Berlin, 1998.
  • S. Todorčević, Combinatorial dichotomies in set theory, Bull. Symbolic Log., 17 (2011), 1–72.
  • B. Veličković, Forcing axioms and stationary sets, Adv. Math., 94 (1992), 256–284.