Skip to main content
SpringerLink
Log in

On Extracting Variable Herbrand Disjunctions

  • Published:
Studia Logica Aims and scope Submit manuscript

Abstract

Some quantitative results obtained by proof mining take the form of Herbrand disjunctions that may depend on additional parameters. We attempt to elucidate this fact through an extension to first-order arithmetic of the proof of Herbrand’s theorem due to Gerhardy and Kohlenbach which uses the functional interpretation.

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. Afshari, B., S. Hetzl, and G. E. Leigh, Herbrand’s theorem as higher order recursion, Annals of Pure and Applied Logic 171(6):102792, 2020.

    Article  Google Scholar 

  2. Ferreira, F., The FAN principle and weak König’s lemma in herbrandized second-order arithmetic, Annals of Pure and Applied Logic 171(9):102843, 2020.

    Article  Google Scholar 

  3. Ferreira, F., Weak König’s lemma in herbrandized classical second-order arithmetic, Portugaliae Mathematica 77,(3–4):399–408, 2020.

    Article  Google Scholar 

  4. Ferreira, F., and G. Ferreira, A herbrandized functional interpretation of classical first-order logic, Archive for Mathematical Logic 56(5–6):523–539, 2017.

    Article  Google Scholar 

  5. Gerhardy, P., and U. Kohlenbach, Extracting Herbrand disjunctions by functional interpretation, Archive for Mathematical Logic 44(5):633–644, 2005.

    Article  Google Scholar 

  6. Gödel, K., Über eine bisher noch nicht benütze Erweiterung des finiten Standpunktes, Dialectica 12:280–287, 1958.

    Article  Google Scholar 

  7. Kohlenbach, U., Herbrand’s theorem and extractive proof theory, SMF – Gazette des Mathématiciens 118:29–41, 2008.

    Google Scholar 

  8. Kohlenbach, U., Applied proof theory: Proof interpretations and their use in mathematics. Springer Monographs in Mathematics, Springer, 2008.

  9. Kohlenbach, U., Proof-theoretic methods in nonlinear analysis. In: Sirakov, B., Ney de Souza, P., Viana, M., (eds.), Proceedings of the International Congress of Mathematicians 2018, vol. 2, World Scientific, 2019, pp. 61–82.

  10. Kohlenbach, U., and L. Leuştean, Asymptotically nonexpansive mappings in uniformly convex hyperbolic spaces, Journal of the European Mathematical Society 12:71–92, 2010.

    Article  Google Scholar 

  11. Kreisel, G., On the interpretation of non-finitist proofs, part I, The Journal Symbolic Logic 16:241–267, 1951.

    Google Scholar 

  12. Kreisel, G., On the interpretation of non-finitist proofs, part II: Interpretation of number theory, applications, The Journal of Symbolic Logic 17:43–58, 1952.

    Article  Google Scholar 

  13. Parsons, C., On \(n\)-quantifier induction, The Journal of Symbolic Logic 37:466–482, 1972.

    Article  Google Scholar 

  14. Shoenfield, J., Mathematical Logic, Addison-Wesley Publishing Co., 1967.

  15. Streicher, T., and U. Kohlenbach, Shoenfield is Gödel after Krivine, Mathematical Logic Quarterly 53(2):176–179, 2007.

    Article  Google Scholar 

  16. Tait, W. W., Infinitely long terms of transfinite type, in: J.N. Crossley, and M.A.E. Dummett, (eds.), Formal Systems and Recursive Functions, vol. 40 of Studies in Logic and the Foundations of Mathematics, Elsevier, Amsterdam, 1965, pp. 176–185.

  17. Tao, T., Soft analysis, hard analysis, and the finite convergence principle; essay posted May 23, 2007; appeared in: Tao, T., Structure and Randomness: Pages from Year One of a Mathematical Blog, American Mathematical Society, 2008.

  18. Tao, T., Norm convergence of multiple ergodic averages for commuting transformations, Ergodic Theory and Dynamical Systems 28:657–688, 2008.

    Article  Google Scholar 

Download references

Acknowledgements

I would like to thank Ulrich Kohlenbach for originally suggesting to me to look at the infinitary calculus which was introduced by Tait in [16], and for the continuing discussions I had with him on the topic. I would also like to thank Pedro Pinto for his suggestions. This work has been supported by a grant of the Romanian Ministry of Research, Innovation and Digitization, CNCS/CCCDI – UEFISCDI, project number PN-III-P1-1.1-PD-2019-0396, within PNCDI III.

Author information

Authors and Affiliations

  1. Research Center for Logic, Optimization and Security (LOS), Department of Computer Science Faculty of Mathematics and Computer Science, University of Bucharest, Academiei 14, 010014, Bucharest, Romania

    Andrei Sipoş

  2. Simion Stoilow Institute of Mathematics of the Romanian Academy, Calea Griviţei 21, 010702, Bucharest, Romania

    Andrei Sipoş

Authors
  1. Andrei Sipoş
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andrei Sipoş.

Additional information

Publisher's Note

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

Presented by Daniele Mundici; Received November 25, 2021.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sipoş, A. On Extracting Variable Herbrand Disjunctions. Stud Logica 110, 1115–1134 (2022). https://doi.org/10.1007/s11225-022-09990-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11225-022-09990-5

Keywords

Mathematics Subject Classification

Search

Navigation