Skip to main content

Transitive Closure and the Mechanization of Mathematics

  • Chapter
Thirty Five Years of Automating Mathematics

Part of the book series: Applied Logic Series ((APLS,volume 28))

Abstract

We argue that the concept of transitive closure is the key for understanding finitary inductive definitions and reasoning, and we provide evidence for the thesis that logics which are based on it (in which induction is a logical rule) are the right logical framework for the formalization and mechanization of mathematics. We investigate the expressive power of languages with the most basic transitive closure operation TC. We show that with TC one can define all recursive predicates and functions from 0, the successor function and addition, yet with TC alone addition is not definable from 0 and the successor function. However, in the presence of a pairing function, TC does suffice for having all types of finitary inductive definitions of relations and functions. This result is used for presenting a simple version of Feferman’s framework FS 0, demonstrating that TC-logics provide in general an excellent framework for mechanizing formal systems. An interesting side effect of these results is a simple characterization of recursive enumerability and a new, concise version of the Church thesis. We end with a use of TC for a formalization of set theory which is based on purely syntactical considerations, and reflects real mathematical practice.

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

Access this chapter

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

eBook
USD 9.99
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
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover 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.

Similar content being viewed by others

Bibliography

  1. A. Avron, F. A. Honsell, I. A. Mason, and R. Pollack. Using Typed Lambda Calculus to Implement Formal Systems on a Machine, Journal of Automated Deduction, 9, 309–354, 1992.

    Article  MathSciNet  MATH  Google Scholar 

  2. A. Avron. Theorems on Strong Constructibility with a Compass alone, Journal of Geometry, 30, 28–35, 1987.

    Article  MathSciNet  MATH  Google Scholar 

  3. A. Avron. On Strict Strong Constructibility with a Compass Alone, Journal of Geometry, 38, 12–15, 1990.

    Article  MathSciNet  MATH  Google Scholar 

  4. A. Avron. An Exercise in An Interactive Geometrical research, Annals of Mathematics and Artificial Intelligence, 9, 239–252, 1993.

    Article  MATH  Google Scholar 

  5. A. Avron. Partial Safety of Formulas as a Unifying Foundational Principle, To appear.

    Google Scholar 

  6. J. Barwise, ed. Handbook of Mathematical Logic, vol. 90 of Studies in Logic and the Foundations of Mathematics, North-Holland, 1977.

    Google Scholar 

  7. J. R. Biichi. On a Decision Method in Restricted Second Order Arithmetic. In Logic Methodology and Philosophy of Science, Proceedings of the 1960 Congress, pp. 1–11. Stanford University Press, Stanford, CA, 1962.

    Google Scholar 

  8. H. D. Ebbinghaus and J. Flum. Finite Model Theory, Perspectives in Mathematical Logic, Springer, 1995.

    MATH  Google Scholar 

  9. S. Feferman. Finitary Inductively Presented Logics/ In Logic Colloquium 1988,pp. 191–220. North-Holland, Amsterdam, 1989. Reprinted in [Gabbay, 1994, pp. 297–328].

    Google Scholar 

  10. D. M. Gabbay, editor. What is a Logical System? Oxford Science Publications, Clarendon Press, Oxford, 1994.

    Google Scholar 

  11. G. Gentzen. Neue Fassung des Widersprtschsfreiheitsbeweises für die reine Zahlentheorie, Forschungen zur Logik, N.S., No. 4, pp. 19–44, 1969. English translation in: The collected work of Gerhard Gentzen, M. E. Szabo, ed. North-Holland, Amsterdam, 1969.

    Google Scholar 

  12. E. Grädel. On Transitive Closure Logic. In Computer Science Logic (Berne 1991), pp. 149–163, vol. 626 of Lecture Notes in Computer Science, Springer, 1992.

    Google Scholar 

  13. Y. Gurevich. Logic and the Challenge of Computer Science. In E. Börger, ed. Trends in Theoretical Computer Science, pp. 1–58. Computer Science Press Inc., Rockville, MD, 1988.

    Google Scholar 

  14. R. Harper, F. Honsell and G. Plotkin. A Framework for Defining Logics, Journal of the Association for Computing Machinery, 40, 143–184, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  15. N. Immerman. Languages which Capture Complexity Classes. In 15th Symposium on Theory of Computing, pp. 347–354. Association for Computing Machinery, 1983.

    Google Scholar 

  16. H. Levesque, R. Reiter, Y. Lesperance, F. Lin, and R. Scherl. Golog: A logic programming language for dynamic domains, Journal of Logic Programming, 31, 59–84, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  17. S. Matthews. A Theory and Its Metatheory in FSv. In [Gabbay, 1994, pp. 329–352].

    Google Scholar 

  18. S. Matthews. Implementing FS0 in Isabelle: Adding Structure at the Metalevel. In Proc. Disco’96, J. Calmet and C. Limongelli, eds. Springer, Berlin, 1996.

    Google Scholar 

  19. Y. Moschovakis. Abstract Recursion as a Foundation for the Theory of Algorithms, pp. 289–364. Vol 1104 of Lecture Notes in Mathematics, Springer, 1984.

    Google Scholar 

  20. S. Matthews, A. Smaill, and D. Basin. Experience with FS0 as a Framework Theory. In Logical Environments, G. Huet and G. Plotkin, eds., pp. 61–82. Cambridge University Press, 1993,.

    Google Scholar 

  21. F. Pfenning. The Practice of Logical Frameworks. In Proceedings of the Colloquium on Trees in Algebra and Programming, Linköping, Sweden, April 1996, H. Kirchner, ed. pp. 119–134. Vol 1059 of Lecture Notes in Computer Science, Springer-Verlag, 1996.

    Google Scholar 

  22. E. Post. Formal Reductions of the General Combinatorial Decision Problem, American J. of Mathematics, 197–214, 1943.

    Google Scholar 

  23. R. Smullyan. Theory of Formal Systems, Princeton University Press, Princeton, 1961.

    MATH  Google Scholar 

  24. A. Tarski and S. Givant. Tarski’s System of Geometry, Bulletin of Symbolic Logic, 5, 175–214, 1999.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2003 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Avron, A. (2003). Transitive Closure and the Mechanization of Mathematics. In: Kamareddine, F.D. (eds) Thirty Five Years of Automating Mathematics. Applied Logic Series, vol 28. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0253-9_7

Download citation

  • DOI: https://doi.org/10.1007/978-94-017-0253-9_7

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-6440-0

  • Online ISBN: 978-94-017-0253-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics