[go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

Categoricity Results and Large Model Constructions for Second-Order ZF in Dependent Type Theory

  • Published:
Journal of Automated Reasoning Aims and scope Submit manuscript

Abstract

We formalise second-order ZF set theory in the dependent type theory of Coq. Assuming excluded middle, we prove Zermelo’s embedding theorem for models, categoricity in all cardinalities, and the categoricity of extended axiomatisations fixing the number of Grothendieck universes. These results are based on an inductive definition of the cumulative hierarchy eliminating the need for ordinals and set-theoretic transfinite recursion. Following Aczel’s sets-as-trees interpretation, we give a concise construction of an intensional model of second-order ZF with a weakened replacement axiom. Whereas this construction depends on no additional logical axioms, we obtain intensional and extensional models with full replacement assuming a description operator for trees and a weak form of proof irrelevance. In fact, these assumptions yield large models with n Grothendieck universes for every number n.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /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.

Institutional subscriptions

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  1. Aczel, P.: The type theoretic interpretation of constructive set theory. Stud. Logic Found. Math. 96, 55–66 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aczel, P.: On Relating Type Theories and Set Theories. Types for Proofs and Programs. Lecture Notes in Computer Science, pp. 1–18. Springer, Berlin (1998)

    Google Scholar 

  3. Barbanera, F., Berardi, S.: Proof-irrelevance out of excluded-middle and choice in the calculus of constructions. J. Funct. Program. 6(3), 519–525 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  4. Barras, B.: Sets in Coq, Coq in sets. J. Formaliz. Reason. 3(1), 29–48 (2010)

    MathSciNet  MATH  Google Scholar 

  5. Bauer, A., Gross, J., Lumsdaine, P.L., Shulman, M., Sozeau, M., Spitters, B.: The HoTT library: a formalization of homotopy type theory in Coq. In: CPP 2017. ACM, New York, NY, USA, pp. 164–172 (2017)

  6. Bourbaki, N.: Sur le théorème de Zorn. Archiv der Math. 2(6), 434–437 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  7. Coquand, T.: Metamathematical investigations of a calculus of constructions (1989)RR-1088, INRIA,<inria-00075471>

  8. Gylterud, H.R.: From multisets to sets in hotmotopy type theory (2016). arXiv: 1612.05468

  9. Hamkins, J.D.: Every countable model of set theory embeds into its own constructible universe. J. Math. Logic 13(02), 1350006 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hrbacek, K., Jech, T.: Introduction to Set Theory, Revised and Expanded, 3rd edn. CRC Press, Boca Raton (1999)

    MATH  Google Scholar 

  11. Kirst, D., Smolka, G.: Categoricity results for second-order ZF in dependent type theory. In: Ayala-Rincón, M., Muñoz, C.A. (eds.) ITP 2017, Brasília, Brazil, September 26–29, 2017, Vol. 10499 of LNCS. Springer, pp. 304–318 (2017)

  12. Kirst, D., Smolka, G.: Large model constructions for second-order ZF in dependent type theory. In: Andronick, J., Felty, A.P. (eds.) CPP 2018, Los Angeles, CA, USA, January 8–9, 2018. ACM, pp. 228–239 (2018)

  13. Kreisel, G.: Two notes on the foundations of set-theory. Dialectica 23(2), 93–114 (1969)

    Article  MATH  Google Scholar 

  14. Kunen, K.: Set Theory: An Introduction to Independence Proofs. Elsevier, New York (2014)

    MATH  Google Scholar 

  15. Ledent, J.: Modeling set theory in homotopy type theory (2014). https://perso.ens-lyon.fr/jeremy.ledent/internship_report.pdf

  16. Paulson, L.C.: Set theory for verification: I. From foundations to functions. J. Autom. Reason. 11(3), 353–389 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  17. Scott, D.: Axiomatizing set theory. Proc. Symp. Pure Math. 13, 207–214 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  18. Skolem, T.: Some remarks on axiomatized set theory. In: van Heijenoort, J. (ed) From Frege to Gödel: A Sourcebook in Mathematical Logic. toExcel, Lincoln, NE, USA, pp. 290–301 (1922)

  19. Smolka, G., Schäfer, S., Doczkal, C.: Transfinite constructions in classical type theory. In: Zhang, X., Urban, C. (eds.) ITP 2015, Nanjing, China, August 24–27, 2015, LNCS 9236. Springer (2015)

  20. Smullyan, R.M., Fitting, M.: Set Theory and the Continuum Problem. Dover Books on Mathematics. Dover Publications, New York (2010)

    Google Scholar 

  21. Sozeau, M., Tabareau, N.: Universe Polymorphism in Coq. Interactive Theorem Proving. Lecture Notes in Computer Science, pp. 499–514. Springer, Cham (2014)

    Book  MATH  Google Scholar 

  22. Suppes, P.: Axiomatic Set Theory. Dover Books on Mathematics Series. Dover Publications, New York (1960)

    Google Scholar 

  23. The Coq Proof Assistant. http://coq.inria.fr (2018)

  24. T. Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics. Institute for Advanced Study. https://homotopytypetheory.org/book (2013)

  25. Uzquiano, G.: Models of second-order Zermelo set theory. Bull. Symb. Logic 5(3), 289–302 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  26. Väänänen, J.: Second-order logic or set theory? Bull. Symb. Logic 18(1), 91–121 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  27. Werner, B.: Sets in types, types in sets. In: Theoretical Aspects of Computer Software. Springer, Heidelberg, pp. 530–546 (1997)

  28. Williams, N.H.: On Grothendieck universes. Compos. Math. 21(1), 1–3 (1969)

    MathSciNet  MATH  Google Scholar 

  29. Zermelo, E.: Neuer Beweis für die Möglichkeit einer Wohlordnung. Math. Ann. 65, 107–128 (1908)

    Article  MATH  Google Scholar 

  30. Zermelo, E.: Über Grenzzahlen und Mengenbereiche: Neue Untersuchungen über die Grundlagen der Mengenlehre. Fundam. Math. 16, 29–47 (1930)

    Article  MATH  Google Scholar 

Download references

Acknowledgements

This research benefited from a quotient construction yielding the extensional model \(\mathcal {S} _i\) first formalised by Chad E. Brown in Coq using a full choice axiom. We also thank the anonymous reviewers for their helpful comments and suggestions that improved the final version of this paper.

Author information

Authors and Affiliations

  1. Saarland University, Saarland Informatics Campus, Saarbrücken, Germany

    Dominik Kirst & Gert Smolka

Authors
  1. Dominik Kirst
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gert Smolka
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dominik Kirst.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kirst, D., Smolka, G. Categoricity Results and Large Model Constructions for Second-Order ZF in Dependent Type Theory. J Autom Reasoning 63, 415–438 (2019). https://doi.org/10.1007/s10817-018-9480-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10817-018-9480-6

Keywords

Search

Navigation