Skip to main content
Log in

Generic Vopěnka’s Principle, remarkable cardinals, and the weak Proper Forcing Axiom

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

Abstract

We introduce and study the first-order Generic Vopěnka’s Principle, which states that for every definable proper class of structures \(\mathcal {C}\) of the same type, there exist \(B\ne A\) in \(\mathcal {C}\) such that B elementarily embeds into A in some set-forcing extension. We show that, for \(n\ge 1\), the Generic Vopěnka’s Principle fragment for \(\Pi _n\)-definable classes is equiconsistent with a proper class of n-remarkable cardinals. The n-remarkable cardinals hierarchy for \(n\in \omega \), which we introduce here, is a natural generic analogue for the \(C^{(n)}\)-extendible cardinals that Bagaria used to calibrate the strength of the first-order Vopěnka’s Principle in Bagaria (Arch Math Logic 51(3–4):213–240, 2012). Expanding on the theme of studying set theoretic properties which assert the existence of elementary embeddings in some set-forcing extension, we introduce and study the weak Proper Forcing Axiom, \(\mathrm{wPFA}\). The axiom \(\mathrm{wPFA}\) states that for every transitive model \(\mathcal M\) in the language of set theory with some \(\omega _1\)-many additional relations, if it is forced by a proper forcing \(\mathbb P\) that \(\mathcal M\) satisfies some \(\Sigma _1\)-property, then V has a transitive model \(\bar{\mathcal M}\), satisfying the same \(\Sigma _1\)-property, and in some set-forcing extension there is an elementary embedding from \(\bar{\mathcal M}\) into \(\mathcal M\). This is a weakening of a formulation of \(\mathrm{PFA}\) due to Claverie and Schindler (J Symb Logic 77(2):475–498, 2012), which asserts that the embedding from \(\bar{\mathcal M}\) to \(\mathcal M\) exists in V. We show that \(\mathrm{wPFA}\) is equiconsistent with a remarkable cardinal. Furthermore, the axiom \(\mathrm{wPFA}\) implies \(\mathrm{PFA}_{\aleph _2}\), the Proper Forcing Axiom for antichains of size at most \(\omega _2\), but it is consistent with \(\square _\kappa \) for all \(\kappa \ge \omega _2\), and therefore does not imply \(\mathrm{PFA}_{\aleph _3}\).

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

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Bagaria, J.: \(C^{(n)}\)-cardinals. Arch. Math. Logic 51(3–4), 213–240 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. Claverie, B., Schindler, R.: Woodin’s axiom \((\ast )\), bounded forcing axioms, and precipitous ideals on \(\omega _1\). J. Symb. Logic 77(2), 475–498 (2012)

    Article  MATH  Google Scholar 

  3. Cheng, Y., Gitman, V.: Indestructibility properties of remarkable cardinals. Arch. Math. Logic 54(7), 961–984 (2015)

  4. Cheng, Y., Schindler, R.: Harrington’s principle in higher order arithmetic. J. Symb. Logic 80(2), 477–489 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  5. Gale, D., Stewart, F.M.: Infinite games with perfect information. Ann. Math. Stud. 28, 245–266 (1953)

    MathSciNet  MATH  Google Scholar 

  6. Gitman, V., Schindler, R.: Virtual large cardinals. In preparation

  7. Gitman, V., Welch, P.D.: Ramsey-like cardinals II. J. Symb. Logic 76(2), 541–560 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hamkins, J.D.: The Vopěnka principle is inequivalent to but conservative over the Vopěnka scheme. Manuscript under review

  9. Kunen, K.: Elementary embeddings and infinitary combinatorics. J. Symb. Logic 36, 407–413 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  10. Magidor, M.: On the role of supercompact and extendible cardinals in logic. Isr. J. Math. 10, 147–157 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  11. Schindler, R.D.: Proper forcing and remarkable cardinals. Bull. Symb. Logic 6(2), 176–184 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  12. Schindler, R.: Remarkable cardinals. In: Infinity, Computability, and Metamathematics: Festschrift in honour of the 60th birthdays of Peter Koepke and Philip Welch, Series: Tributes. College Publications, London, GB (2014)

  13. Schindler, R.: Set theory. Universitext. Springer, Cham (2014). Exploring independence and truth

  14. Todorčević, S.: A note on the proper forcing axiom. In: Axiomatic Set Theory (Boulder, CO, 1983), vol. 31 of Contemp. Math., pp. 209–218. Am. Math. Soc., Providence, RI (1984)

  15. Todorčević, S.: Localized reflection and fragments of PFA. In: Set Theory (Piscataway, NJ, 1999), vol. 58 of DIMACS Ser. Discrete Math. Theoret. Comput. Sci., pp. 135–148. Am. Math. Soc., Providence, RI (2002)

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Victoria Gitman.

Additional information

Parts of this research were done while all three authors were visiting fellows at the Isaac Newton Institute for Mathematical Sciences, Cambridge, UK, in the programme “Mathematical, Foundational and Computational Aspects of the Higher Infinite” (HIF) funded by EPSRC Grant EP/K032208/1 in September 2015. The first author would like to thank the support provided by a Simons Foundation fellowship while at the INI. Other parts of this research were done while the third author was visiting the School of Mathematics of the IPM, Tehran, Iran, in October 2015; he would like to thank his hosts for their exceptional hospitality. The research work of the first author was partially supported by the Spanish Government under Grant MTM2014-59178-P, and by the Generalitat de Catalunya (Catalan Government) under Grant SGR 437-2014.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bagaria, J., Gitman, V. & Schindler, R. Generic Vopěnka’s Principle, remarkable cardinals, and the weak Proper Forcing Axiom. Arch. Math. Logic 56, 1–20 (2017). https://doi.org/10.1007/s00153-016-0511-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00153-016-0511-x

Keywords

Mathematics Subject Classification

Navigation