Skip to main content
SpringerLink
Log in
Book cover

International Symposium on Logical Foundations of Computer Science

LFCS 2020: Logical Foundations of Computer Science pp 249–267Cite as

Lifting Recursive Counterexamples to Higher-Order Arithmetic

  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11972))

Abstract

In classical computability theory, a recursive counterexample to a theorem shows that the latter does not hold when restricted to computable objects. These counterexamples are highly useful in the Reverse Mathematics program, where the aim of the latter is to determine the minimal axioms needed to prove a given theorem of ordinary mathematics. Indeed, recursive counterexamples often (help) establish the ‘reverse’ implication in the typical equivalence between said minimal axioms and the theorem at hand. The aforementioned is generally formulated in the language of second-order arithmetic and we show in this paper that recursive counterexamples are readily modified to provide similar implications in higher-order arithmetic. For instance, the higher-order analogue of ‘sequence’ is the topological notion of ‘net’, also known as ‘Moore-Smith sequence’. Our results on metric spaces suggest that the latter can only be reasonably studied in weak systems via representations (aka codes) in the language of second-order arithmetic.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    To be absolutely clear, variables (of any finite type) are allowed in quantifier-free formulas of the language \(\textsf {L }_{\omega }\): only quantifiers are banned.

  2. 2.

    An equivalence relation is a binary, reflexive, transitive, and symmetric relation.

References

  1. Avigad, J., Feferman, S.: Gödel’s functional (“dialectica”) interpretation. In: Handbook of Proof Theory. Studies in Logic and the Foundations of Mathematics, vol. 137, pp. 337–405 (1998)

    Google Scholar 

  2. Bartle, R.G.: Nets and filters in topology. Am. Math. Mon. 62, 551–557 (1955)

    Article  MathSciNet  Google Scholar 

  3. Brown, D.K.: Notions of closed subsets of a complete separable metric space in weak subsystems of second-order arithmetic. In: Logic and Computation, Pittsburgh, PA (1987). Contemporary Mathematics, vol. 106, pp. 39–50. American Mathematical Society, Providence, RI (1990)

    Google Scholar 

  4. Brown, D.K.: Notions of compactness in weak subsystems of second order arithmetic. In: Simpson, S. (ed.) Reverse Mathematics 2001. Perspectives in Logic, pp. 47–66. Association for Symbolic Logic, Urbana, IL (2005)

    Google Scholar 

  5. Dorais, F.G., Hirst, J.L., Shafer, P.: Reverse mathematics, trichotomy and dichotomy. J. Log. Anal. 4, 14 (2012). Paper 13

    MathSciNet  MATH  Google Scholar 

  6. Ershov, Y.L.: Theorie der numerierungen III. Z. Math. Logik Grundlagen Math. 23(4), 289–371 (1977)

    MathSciNet  MATH  Google Scholar 

  7. Ershov, Y.L., Goncharov, S.S., Nerode, A., Remmel, J.B., Marek, V.W. (eds.): Handbook of Recursive Mathematics. Volume 138 of Studies in Logic and the Foundations of Mathematics, Vol. 1. North-Holland, Amsterdam (1998). Recursive Model Theory

    Google Scholar 

  8. Friedman, H.: Some systems of second order arithmetic and their use. In: Proceedings of the International Congress of Mathematicians (Vancouver, B.C., 1974), vol. 1, pp. 235–242 (1975)

    Google Scholar 

  9. Friedman, H.: Systems of second order arithmetic with restricted induction, I & II (abstracts). J. Symb. Log. 41, 557–559 (1976)

    Article  Google Scholar 

  10. Friedman, H., Simpson, S.G., Smith, R.: Countable algebra and set existence axioms. Ann. Pure Appl. Logic 25(2), 141–181 (1983)

    Article  MathSciNet  Google Scholar 

  11. Hatzikiriakou, K.: Minimal prime ideals and arithmetic comprehension. J. Symb. Log. 56(1), 67–70 (1991)

    Article  MathSciNet  Google Scholar 

  12. Hirst, J.L.: Combinatorics in subsystems of second order arithmetic, Ph.D. thesis, The Pennsylvania State University, ProQuest LLC, Ann Arbor, MI (1987)

    Google Scholar 

  13. Hunter, J.: Higher-order reverse topology, Ph.D. thesis, The University of Wisconsin-Madison, ProQuest LLC, Ann Arbor, MI (2008)

    Google Scholar 

  14. Kelley, J.L.: General Topology. Springer, New York (1975). Reprint of the 1955 edition; Graduate Texts in Mathematics, no. 27

    MATH  Google Scholar 

  15. Kohlenbach, U.: Foundational and mathematical uses of higher types. In: Reflections on the Foundations of Mathematics (Stanford, CA, 1998). Lecture Notes in Logic 15, pp. 92–116. Association for Symbolic Logic, Urbana, IL (2002)

    Google Scholar 

  16. Kohlenbach, U.: Higher order reverse mathematics. In: Simpson, S. (ed.) Reverse Mathematics 2001. Perspectives in Logic, pp. 281–295. Association for Symbolic Logic, Urbana, IL (2005)

    Google Scholar 

  17. Mandelkern, M.: Brouwerian counterexamples. Math. Mag. 62(1), 3–27 (1989)

    Article  MathSciNet  Google Scholar 

  18. Moore, E.H., Smith, H.: A general theory of limits. Am. J. Math. 44, 102–121 (1922)

    Article  MathSciNet  Google Scholar 

  19. Normann, D., Sanders, S.: On the mathematical and foundational significance of the uncountable. J. Math. Log. (2018). https://doi.org/10.1142/S0219061319500016

    Article  MathSciNet  Google Scholar 

  20. Normann, D., Sanders, S.: Pincherle’s theorem in Reverse Mathematics and computability theory. arXiv:1808.09783 (2018, submitted)

  21. Rado, R.: Axiomatic treatment of rank in infinite sets. Can. J. Math. 1, 337–343 (1949)

    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. Sanders, S.: Plato and the foundations of mathematics. arxiv:1908.05676, pp. 44 (2019, submitted)

  25. Sanders, S.: Nets and reverse mathematics: a pilot study, to appear in Computability, pp. 34 (2019)

    Google Scholar 

  26. Sanders, S.: Nets and reverse mathematics. In: Manea, F., Martin, B., Paulusma, D., Primiero, G. (eds.) CiE 2019. LNCS, vol. 11558, pp. 253–264. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-22996-2_22

    Chapter  Google Scholar 

  27. Sanders, S.: Reverse mathematics and computability theory of domain theory. In: Iemhoff, R., Moortgat, M., de Queiroz, R. (eds.) WoLLIC 2019. LNCS, vol. 11541, pp. 550–568. Springer, Heidelberg (2019). https://doi.org/10.1007/978-3-662-59533-6_33

    Chapter  Google Scholar 

  28. Simpson, S.G.: Subsystems of Second Order Arithmetic. Perspectives in Logic, 2nd edn. Cambridge University Press, Cambridge (2009)

    Book  Google Scholar 

  29. Simpson, S.G., Rao, J.: Reverse algebra. In: Handbook of Recursive Mathematics, Volume 2, Studies in Logic and the Foundations of Mathematics, vol. 139, pp. 1355–1372. North-Holland (1998)

    Google Scholar 

  30. Solomon, D.R.: Reverse mathematics and ordered groups, Ph.D. thesis, Cornell University, ProQuest LLC (1998)

    Google Scholar 

  31. Specker, E.: Nicht konstruktiv beweisbare Sätze der Analysis. J. Symb. Log. 14, 145–158 (1949)

    Article  Google Scholar 

  32. Stillwell, J.: Reverse Mathematics, Proofs from the Inside Out. Princeton University Press, Princeton (2018)

    Book  Google Scholar 

  33. Troelstra, A.S.: Metamathematical Investigation of Intuitionistic Arithmetic and Analysis. Lecture Notes in Mathematics, vol. 344. Springer, Berlin (1973). https://doi.org/10.1007/BFb0066739

    Book  MATH  Google Scholar 

  34. Turing, A.: On computable numbers, with an application to the Entscheidungs-problem. Proc. Lond. Math. Soc. 42, 230–265 (1936)

    MATH  Google Scholar 

  35. Vietoris, L.: Stetige Mengen (German). Monatsh. Math. Phys. 31, 173–204 (1921)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgement

Our research was supported by the John Templeton Foundation via the grant a new dawn of intuitionism with ID 60842. We express our gratitude towards this institution. We thank Anil Nerode and Paul Shafer for their valuable advice. We also thank the anonymous referees for the helpful suggestions. Opinions expressed in this paper do not necessarily reflect those of the John Templeton Foundation.

Author information

Authors and Affiliations

  1. Department of Mathematics, TU Darmstadt, Darmstadt, Germany

    Sam Sanders

Authors
  1. Sam Sanders
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sam Sanders .

Editor information

Editors and Affiliations

  1. The Graduate Center, CUNY, New York, NY, USA

    Sergei Artemov

  2. Cornell University, Ithaca, NY, USA

    Anil Nerode

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Sanders, S. (2020). Lifting Recursive Counterexamples to Higher-Order Arithmetic. In: Artemov, S., Nerode, A. (eds) Logical Foundations of Computer Science. LFCS 2020. Lecture Notes in Computer Science(), vol 11972. Springer, Cham. https://doi.org/10.1007/978-3-030-36755-8_16

Download citation

Publish with us

Policies and ethics

Search

Navigation