Skip to main content
SpringerLink
Log in

Simple proofs of \({\mathsf{SCH}}\) from reflection principles without using better scales

  • Published:
Archive for Mathematical Logic Aims and scope Submit manuscript

Abstract

We give simple proofs of the Singular Cardinal Hypothesis from the Weak Reflection Principle and the Fodor-type Reflection Principle which do not use better scales.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Doebler P., Schindler R.: \({\Pi_2}\) consequences of \({{\rm BMM} + {\rm NS}_{\omega_1}}\) is precipitous and the semiproperness of stationary set preserving forcings. Math. Res. Lett. 16(5), 797–815 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Feng Q., Magidor M.: On reflection of stationary sets. Fund. Math. 140(2), 175–181 (1992)

    MathSciNet  MATH  Google Scholar 

  3. Foreman M., Magidor M., Shelah S.: Martin’s maximum, saturated ideals, and nonregular ultrafilters I. Ann. Math. (2) 127(1), 1–47 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  4. Fuchino S., Juhász I., Soukup L., Szentmiklóossy Z., Usuba T.: Fodor-type reflection principle and reflection of metrizability and meta-Lindelöfness. Topol. Appl. 157(8), 1415–1429 (2010)

    Article  MATH  Google Scholar 

  5. Fuchino S., Rinot A.: Openly generated Boolean algebras and the Fodor-type reflection principle. Fund. Math. 212(3), 261–283 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Fuchino S., Soukup L., Sakai H., Usuba T.: More on Fodor-type Reflection Principle. preprint

  7. Sakai H., Veličković B.: Stationary reflection principles and two cardinal tree properties. J. Inst. Math. Jussieu 14(1), 69–85 (2015)

    Article  MathSciNet  Google Scholar 

  8. Shelah S.: Cardinal Arithmetic. Oxford Logic Guides, vol. 29. Oxford University Press, London (1994)

    Google Scholar 

  9. Shelah S.: Proper and Improper Forcing, 2nd edn. Perspectives in Mathematical Logic. Springer, Berlin (1998)

  10. Shelah S.: Reflection implies the SCH. Fund. Math. 198(2), 95–111 (2008)

    Article  MathSciNet  Google Scholar 

  11. Shelah S., Shioya M.: Nonreflecting stationary sets in \({\mathcal{P}_\kappa \lambda}\). Adv. Math. 199(1), 185–191 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Silver J.: On the singular cardinals problem. In: Proceedings of the International Congress of Mathematicians (Vancouver, B.C., 1974), vol. 1, pp. 265–268, Canadian Mathematical Congress, Montreal, Quebec (1975)

  13. Solovay R.M.: Strongly compact cardinals and the GCH. In: Proceedings of the Tarski Symposium (Proceedings of Symposium on Pure Mathematical, vol. XXV, Univ. California, Berkeley, CA, 1971), pp. 365–372, American Mathematical Society, Providence, RI (1974)

  14. Viale M.: The proper forcing axiom and the singular cardinal hypothesis. J. Symb. Logic 71(2), 473–479 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. Viale M., Weiß C.: On the consistency strength of the proper forcing axiom. Adv. Math. 228(5), 2672–2687 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Veličković B.: Forcing axioms and stationary sets. Adv. Math. 94(2), 256–284 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  17. Weiß C.: The combinatorial essence of supercompactness. Ann. Pure Appl. Logic 163(11), 1710–1717 (2012)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Graduate School of System Informatics, Kobe University, 1-1 Rokkodai-cho, Nada, Kobe, 657-8501, Japan

    Hiroshi Sakai

Authors
  1. Hiroshi Sakai
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hiroshi Sakai.

Additional information

This research was supported by JSPS and FWF under the Japan—Austria Research Cooperative Program “New Developments regarding Forcing in Set Theory”.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sakai, H. Simple proofs of \({\mathsf{SCH}}\) from reflection principles without using better scales. Arch. Math. Logic 54, 639–647 (2015). https://doi.org/10.1007/s00153-015-0432-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00153-015-0432-0

Keywords

Mathematics Subject Classification

Search

Navigation