Skip to main content
SpringerLink
Log in

Equivalence of generics

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

Abstract

Given a countable transitive model of set theory and a partial order contained in it, there is a natural countable Borel equivalence relation on generic filters over the model; two are equivalent if they yield the same generic extension. We examine the complexity of this equivalence relation for various partial orders, focusing on Cohen and random forcing. We prove, among other results, that the former is an increasing union of countably many hyperfinite Borel equivalence relations, and hence is amenable, while the latter is neither amenable nor treeable.

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

Data Availibility Statement

All data relevant to the present work is contained herein

Notes

  1. We do not really need M to be countable; for a given partial order \({\mathbb {P}}\), it suffices that \({\mathcal {P}}({\mathbb {P}})\cap M\) is countable in \({\mathbf {V}}\). In particular, we could allow M to be a transitive class in \({\mathbf {V}}\), such as when \({\mathbf {V}}\) is a generic extension of M after sufficient collapsing, or \(M={\mathbf {L}}\) under large cardinal hypotheses. In such cases, only the proofs of Lemmas 2.6 and 2.9 need alteration, instead relying on Theorem 2.16.

  2. Being hyperfinite is a \(\Sigma ^1_2\) property in the codes, but whether it is \(\Sigma ^1_2\)-complete, and thus not absolute for countable models, appears to be unknown.

  3. We would like to thank the anonymous referee for clarification on this point.

  4. As per footnote 1, we may allow \(\omega _1\subseteq M\), provided \(|{\mathcal {P}}({\mathbb {B}})\cap M|=|{\mathcal {P}}({\mathbb {R}})\cap M|\) is still countable in \({\mathbf {V}}\). In this case, M will be \(\mathbf {\Sigma }^1_2\)-correct by Shoenfield absoluteness (Theorem 25.20 in [13]).

References

  1. Adams, S.R., Spatzier, R.J.: Kazhdan groups, cocycles and trees. Am. J. Math. 112(2), 271–287 (1990)

    Article  MathSciNet  Google Scholar 

  2. Clemens, J., Coskey, S., Dworetzky, S.: The classification of countable models of set theory. MLQ Math. Log. Q. 66(2), 182–189 (2020)

    Article  MathSciNet  Google Scholar 

  3. Dobrinen, N., Friedman, S.-D.: Homogeneous iteration and measure one covering relative to HOD. Arch. Math. Logic 47(7–8), 711–718 (2008)

    Article  MathSciNet  Google Scholar 

  4. Dougherty, R., Jackson, S., Kechris, A.S.: The structure of hyperfinite Borel equivalence relations. Trans. Am. Math. Soc. 341(1), 193–225 (1994)

    Article  MathSciNet  Google Scholar 

  5. Feldman, J., Moore, C.C.: Ergodic equivalence relations, cohomology, and von Neumann algebras. I. Trans. Am. Math. Soc. 234(2), 289–324 (1977)

    Article  MathSciNet  Google Scholar 

  6. Gao, S.: Invariant Descriptive Set Theory. Pure and Applied Mathematics, vol. 293. CRC Press, Boca Raton (2009)

  7. Grigorieff, S.: Intermediate submodels and generic extensions in set theory. Ann. Math. 2(101), 447–490 (1975)

    Article  MathSciNet  Google Scholar 

  8. Habič, M.E., Hamkins, J.D., Klausner, L.D., Verner, J., Williams, K.J.: Set-theoretic blockchains. Arch. Math. Logic 58(7–8), 965–997 (2019)

    Article  MathSciNet  Google Scholar 

  9. Hamkins, J.D.: On \({V}\)-decisive and weakly homogeneous forcings. MathOverflow. https://mathoverflow.net/q/184806 (version: 2017-04-13)

  10. Hayut, Y., Karagila, A.: Restrictions on forcings that change cofinalities. Arch. Math. Logic 55(3–4), 373–384 (2016)

    Article  MathSciNet  Google Scholar 

  11. Hjorth, G., Kechris, A.S.: Borel equivalence relations and classifications of countable models. Ann. Pure Appl. Logic 82(3), 221–272 (1996)

    Article  MathSciNet  Google Scholar 

  12. Jackson, S., Kechris, A.S., Louveau, A.: Countable Borel equivalence relations. J. Math. Log. 2(1), 1–80 (2002)

    Article  MathSciNet  Google Scholar 

  13. Jech, T.: Set Theory. Springer Monographs in Mathematics. Springer, Berlin (2003). The Third Millennium Edition, Revised and Expanded

  14. Jech, T., Shelah, S.: A complete Boolean algebra that has no proper atomless complete subalgebra. J. Algebra 182(3), 748–755 (1996)

    Article  MathSciNet  Google Scholar 

  15. Kechris, A.S., Miller, B.D.: Topics in Orbit Equivalence. Lecture Notes in Mathematics, vol. 1852. Springer, Berlin (2004)

    Book  Google Scholar 

  16. Kunen, K.: Set Theory. Studies in Logic and the Foundations of Mathematics, vol. 102. North-Holland, Amsterdam (1980)

  17. Maharam, D., Stone, A.H.: Realizing isomorphisms of category algebras. Bull. Austral. Math. Soc. 19(1), 5–10 (1978)

    Article  MathSciNet  Google Scholar 

  18. Marks, A.S.: Uniformity, universality, and computability theory. J. Math. Log. 17(1), 1750003 (2017)

  19. McAloon, K.: Consistency results about ordinal definability. Ann. Math. Logic 2(4), 449–467 (1970/1971)

  20. Moschovakis, Y.N.: Descriptive Set Theory, Volume 155 of Mathematical Surveys and Monographs, 2nd edn. American Mathematical Society, Providence (2009)

  21. Popa, S.: Cocycle and orbit equivalence superrigidity for malleable actions of \(w\)-rigid groups. Invent. Math. 170(2), 243–295 (2007)

    Article  MathSciNet  Google Scholar 

  22. Royden, H.L.: Real Analysis, 3rd edn. Macmillan Publishing Company, New York (1988)

    MATH  Google Scholar 

  23. Rudin, W.: Real and Complex Analysis, 3rd edn. McGraw-Hill, New York (1987)

    MATH  Google Scholar 

  24. Slaman, T.A., Steel, J.R.: Definable functions on degrees. In: Cabal Seminar 81–85, Volume 1333 of Lecture Notes in Mathematics, pp. 37–55. Springer, Berlin (1988)

  25. Solovay, R.M.: A model of set-theory in which every set of reals is Lebesgue measurable. Ann. Math. 2(92), 1–56 (1970)

    Article  MathSciNet  Google Scholar 

  26. Sullivan, D., Weiss, B., Wright, J.D.M.: Generic dynamics and monotone complete \(C^\ast \)-algebras. Trans. Am. Math. Soc. 295(2), 795–809 (1986)

    MathSciNet  MATH  Google Scholar 

  27. Thomas, S.: Martin’s conjecture and strong ergodicity. Arch. Math. Logic 48(8), 749–759 (2009)

  28. Thomas, S.: Popa superrigidity and countable Borel equivalence relations. Ann. Pure Appl. Logic 158(3), 175–189 (2009)

    Article  MathSciNet  Google Scholar 

  29. Thomas, S., Schneider, S.: Countable Borel equivalence relations. In: Cummings, J., Schimmerling, E. (Eds.) Appalachian Set Theory 2006–2012, Volume 406 of Lecture Notes Series, pp. 25–62. London Mathematical Society (2012)

  30. von Neumann, J.: Einige Sätze über messbare Abbildungen. Ann. Math. (2) 33(3), 574–586 (1932)

  31. Vopěnka, P., Hájek, P.: The Theory of Semisets. Academia (Publishing House of the Czechoslovak Academy of Sciences), Prague (1972)

    MATH  Google Scholar 

  32. Zapletal, J.: Forcing Idealized. Cambridge Tracts in Mathematics, vol. 174. Cambridge University Press, Cambridge (2008)

  33. Zimmer, R.J.: Ergodic Theory and Semisimple Groups. Monographs in Mathematics, vol. 81. Birkhäuser, Basel (1984)

Download references

Acknowledgements

I would like to thank Samuel Coskey, Joel David Hamkins, Andrew Marks, and Simon Thomas for many helpful conversations and correspondences that contributed to this work.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI, 48109, USA

    Iian B. Smythe

Authors
  1. Iian B. Smythe
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Iian B. Smythe.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Smythe, I.B. Equivalence of generics. Arch. Math. Logic 61, 795–812 (2022). https://doi.org/10.1007/s00153-021-00813-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00153-021-00813-3

Keywords

Mathematics Subject Classification

Search

Navigation