Skip to main content
SpringerLink
Log in

A Theory of Explicit Mathematics Equivalent to ID 1

  • Conference paper
  • First Online:
Computer Science Logic (CSL 2000)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1862))

Included in the following conference series:

Abstract

We show that the addition of name induction to the theory \( EETJ + \left( {\mathcal{L}_{EM} - I_N } \right) \) of explicit elementary types with join yields a theory proof-theoretically equivalent to ID 1.

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

Access this chapter

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 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
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

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Martín Abadi and Luca Cardelli. A Theory of Objects. Springer, 1996.

    Google Scholar 

  2. Michael Beeson. Foundations of Constructive Mathematics. Ergebnisse der Mathematik und ihrer Grenzgebiete; 3.Folge, Bd. 6. Springer, Berlin, 1985.

    MATH  Google Scholar 

  3. Wilfried Buchholz, Solomon Feferman, Wolfram Pohlers, and Wilfried Sieg. Iterated Inductive Definitions and Subsystems of Analysis: Recent Proof-Theoretical studies, volume 897 of Lecture Notes in Mathematics. Springer-Verlag, 1981.

    Google Scholar 

  4. Andrea Cantini. Logical Frameworks for Truth and Abstraction, volume 135 of Studies in Logic and the Foundations of Mathematics.North-Holland, 1996.

    Google Scholar 

  5. Solomon Feferman. Formal theories for transfinite iterations of generalized inductive definitions and some subsystems of analysis. In A. Kino, J. Myhill, and R. Vesley, editors, Intuitionismus and Proof Theory, pages 303–326. North Holland, Amsterdam, 1970.

    Google Scholar 

  6. Solomon Feferman. A language and axioms for explicit mathematics. In J. Crossley, editor, Algebra and Logic, volume 450 of Lecture Notes in Mathematics, pages 87–139. Springer, 1975.

    Google Scholar 

  7. Solomon Feferman. Constructive theories of functions and classes. In M. Boffa, D. van Dalen, and K. McAloon, editors, Logic Colloquium 78, pages 159–224. North-Holland, Amsterdam, 1979.

    Google Scholar 

  8. Solomon Feferman. Hilbert’s program relativized: Proof-theoretical and foundational reductions. Journal of Symbolic Logic, 53(2):364–384, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  9. Solomon Feferman. Polymorphic typed lambda-calculus in a type-free axiomatic framework. In W. Sieg, editor, Logic and Computation, volume 106 of Contemporary Mathematics, pages 101–136. American Mathematical Society, 1990.

    Google Scholar 

  10. Solomon Feferman. Logics for termination and correctness of functional programs. In Y. Moschovakis, editor, Logic from Computer Sciences, pages 95–127. Springer, 1991.

    Google Scholar 

  11. Solomon Feferman. Logics for termination and correctness of functional programs II: Logics of strength PRA. In P. Aczel, H. Simmons, and S.S. Wainer, editors, Proof Theory, pages 195–225. Cambridge University Press, 1992.

    Google Scholar 

  12. Solomon Feferman. Does reductive proof theory have a viable rationale? Erkenntnis, 200x.To appear.

    Google Scholar 

  13. Solomon Feferman and Gerhard Jäger. Systems of explicit mathematics with non-constructive μ-operator.P art I. Annals of Pure and Applied Logic, 65(3):243–263, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  14. Solomon Feferman and Gerhard Jäger. Systems of explicit mathematics with non-constructive μ-operator. Part II. Annals of Pure and Applied Logic, 79:37–52, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  15. Ed Griffor and Michael Rathjen. The strength of some Martin-Löf type theories. Archive for Mathematical Logic, 33:347–385, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  16. Gerhard Jäger. Induction in the elementary theory of types and names. In E. Börger, H. Kleine Büning, and M.M. Richter, editors, Computer Science Logic’ 87, volume 329 of Lecture Notes in Computer Science, pages 118–128. Springer, 1988.

    Google Scholar 

  17. Gerhard Jäger, Reinhard Kahle, and Thomas Strahm. On applicative theories. In A. Cantini, E. Casari, and P. Minari, editors, Logic and Foundation of Mathematics, pages 83–92. Kluwer, 1999.

    Google Scholar 

  18. Gerhard Jäger, Reinhard Kahle, and Thomas Studer. Universes in explicit mathematics.200x.Submitted.

    Google Scholar 

  19. Georg Kreisel. Generalized inductive definitions. T echnical report, Stanford Report, 1963.

    Google Scholar 

  20. Markus Marzetta. Predicative Theories of Types and Names. Dissertation, Universität Bern, Institut für Informatik und angewandte Mathematik, 1994.

    Google Scholar 

  21. Markus Marzetta and Thomas Strahm. The μ quantification operator in explicit mathematics with universes and iterated fixed point theories with ordinals. Archive for Mathematical Logic, 37:391–413, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  22. Erik Palmgren. On universes in type theory. In G. Sambin and J. Smith, editors, Twenty Five Years of Constructive Type Theory, pages 191–204. Oxford University Press, 1998.

    Google Scholar 

  23. Wolfram Pohlers. Proof Theory, volume 1407 of Lecture Notes in Mathematics. Springer, 1989.

    Google Scholar 

  24. Robert Stärk. Call-by-value, call-by-name and the logic of values.In D. van Dalen and M. Bezem, editors, Computer Science Logic CSL’ 96: Selected Papers, volume 1258 of Lecture Notes in Computer Science, pages 431–445. Springer, 1997.

    Google Scholar 

  25. Robert Stärk. Why the constant ‘unde.ned’? Logics of partial terms for strict and non-strict functional programming languages. Journal of Functional Programming, 8(2):97–129, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  26. Thomas Studer. A semantics for λstr 1 a calculus with overloading and latebinding. Journal of Logic and Computation, 200x.T o appear.

    Google Scholar 

  27. Anne Troelstra and Dirk van Dalen. Constructivism in Mathematics, volume II.North Holland, Amsterdam, 1988.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. WSI, Universität Tübingen, Sand 13, D-72076, Tübingen, Germany

    Reinhard Kahle

  2. IAM, Universität Bern, Neubrückstr. 10, CH-3012, Bern, Switzerland

    Thomas Studer

Authors
  1. Reinhard Kahle
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Thomas Studer
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer Science and Department of Biology, Boston College, Fulton Hall 410B, Chestnut Hill, MA, 02467, USA

    Peter G. Clote

  2. Mathematisches Institut, Ludwig-Maximilian-Universität München, Theresienstr. 39, 80333, München, Germany

    Helmut Schwichtenberg

Rights and permissions

Reprints and permissions

Copyright information

© 2000 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Kahle, R., Studer, T. (2000). A Theory of Explicit Mathematics Equivalent to ID 1 . In: Clote, P.G., Schwichtenberg, H. (eds) Computer Science Logic. CSL 2000. Lecture Notes in Computer Science, vol 1862. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44622-2_24

Download citation

  • DOI: https://doi.org/10.1007/3-540-44622-2_24

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-67895-3

  • Online ISBN: 978-3-540-44622-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation