Skip to main content
Log in

On some formalized conservation results in arithmetic

  • Published:
Archive for Mathematical Logic Aims and scope Submit manuscript

Abstract

n and n are well known fragments of first-order arithmetic with induction and collection forΣ n formulas respectively; 0n and 0n are their second-order counterparts. RCA0 is the well known fragment of second-order arithmetic with recursive comprehension;WKL 0 isRCA 0 plus weak König's lemma. We first strengthen Harrington's conservation result by showing thatWKL 0 + 0 n is Π 11 -conservative overRCA 0 + 0 n . Then we develop some model theory inWKL 0 and illustrate the use of formalized model theory by giving a relatively simple proof of the fact that 1 proves n+1 to be Π n+2-conservative over n . Finally, we present a proof-theoretic proof of the stronger fact that theΠ n+2 conservation result is provable already in 0 + superexp. Thus n+1 proves 1-Con ( n+1) and 0 +superexp proves Con ( n )↔Con( n+1).

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

Access this article

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.

Similar content being viewed by others

References

  1. Bell, J., Machover, M.: A course in mathematical, logic, 2nd edn. Amsterdam New York Oxford Tokyo: North-Holland 1988

    Google Scholar 

  2. Buss, S.R.: Bounded arithmetic. Studies in proof theory. Napoli: Bibliopolis 1986

    Google Scholar 

  3. Clote, P.G.: Partition relations in arithmetic. In: DiPrisco, C.A. (ed.) Methods in mathematical logic. Proceedings, Caracas 1983. (Lect Notes Math., vol. 1130, pp. 32–68. Berlin Heidelberg New York:Springer 1985

    Google Scholar 

  4. Clote, P.G.: Applications of the low basis theorem in arithmetic. In: Ebbinghaus, H.-D., Müller, G.H., Sacks, G.E. (eds.) Recursion theory week. Proceedings, Oberwolfach, 1984. (Lect. Notes Math., vol. 1141, pp. 65–88). Berlin Heidelberg New York: Springer 1985

    Google Scholar 

  5. Clote, P.G.: Ultrafilters on definable sets in arithmetic. In: Paris, J.B., Wilkie, A.J., Wilmers, G.M. (eds.) Methods in mathematical logic. Logic Colloquium '84, pp. 37–58. Amsterdam New York Oxford Tokyo: North-Holland 1986

    Google Scholar 

  6. Dimitracopoulos, C.: Matijasevic's theorem and fragments of arithmetic. Ph.D. thesis, University of Manchester, 1980

  7. Friedman, H.M.: Some systems of second-order arithmetic and their use. Proceedings of the International Congress of Mathematicians in Vacounver 1974, pp. 235–242 (1975)

  8. Friedman, H.M.: Systems of second-order arithmetic with limited induction (Abstract). J. Symb. Logic41, 557–559 (1976)

    Google Scholar 

  9. Gaifman, H., Dimitracopoulos, C.: Fragments of Peano's arithmetic and the MRDP theorem. In: Logic and Algorithm. Monogr. No. 30, l'Enseignement Mathématique, Université de Genève, Geneva, pp. 187–206 (1982)

    Google Scholar 

  10. Girard, J.-Y.: Proof theory and logical complexity, vol. 1, p. 503. Napoli: Bibliopolis 1987, 503 pages

    Google Scholar 

  11. Hájek, P., Paris, J.B.: Combinatorial principles concerning approximation of functions. Arch. Math. Logic,26, 13–28 (1986)

    Google Scholar 

  12. Jockusch, C.G., Jr., Soare, R.I.: Π 01 -classes and degrees of theories. Trans. Am. Math. Soc.173, 33–56 (1972)

    Google Scholar 

  13. Hájek, P., Kučera, A.: On recursion theory in IΣ1. J. Symb. Logic,54, 576–589 (1989)

    Google Scholar 

  14. Paris, J.B.: A hierarchy of cuts in models of airthmetic. In: Pacholski, L., Wierzejewski, J., Wilkie, A.J. (eds.). Model theory of algebra and arithmetic. Proceedings, Karpacz, Poland 1979. (Lect. Notes Math., vol. 834, pp. 312–337). Berlin Heidelberg New York: Springer 1979

    Google Scholar 

  15. Paris, J.B.: Some conservation results for fragments of arithmetic. In: Berline, C., McAloon, K., Ressayre, J.-P. (eds.) Model theory and arithmetic. Proceedings, Paris 1979/80. (Lect. Notes Math., vol. 890, pp. 251–262). Berlin Heidelberg New York: Springer 1981

    Google Scholar 

  16. Paris, J., Kirby, L.:Σ n -collection schemas in arithmetic. In: Logic Colloquium '77, pp. 199–209. Amsterdam New York Oxford Tokyo: North-Holland 1978

    Google Scholar 

  17. Robinson, R.M.: An essentially undecidable axiom system. Proc. Int. Cong. Math., vol. I, pp. 729–730. Cambridge, MA 1950

    Google Scholar 

  18. Shoenfield, J.R.: Mathematical Logic, 2nd edn. Addison-Wesley 1967

  19. Sieg, W.: Provably recursive functionals of theories with König's lemma, (to appear in Rend. Sem. Mat. Univers. Politecn. Torino, Fasc. Speciale Logic & Comp. Sc.: some of the results of this article were announed in the Logic Colloquium '85 in Orsay, France)

  20. Takeuti, G.: Proof. theory, 2nd edn., vol. 81. Studies in logic and the foundation of mathematics. Amsterdam New York Oxford Tokyo: North-Holland 1987

    Google Scholar 

  21. Wilkie, A.J., Paris, J.B.: On the scheme of induction for bounded formulas in arithmetic. Ann. Pure Appl. Logic35, 261–302 (1987)

    Google Scholar 

  22. Woods, A.: Some problems in logic and number theory. Ph.D. thesis, University of Manchester (1981)

Download references

Author information

Authors and Affiliations

Authors

Additional information

The first author was partially supported by NSF Grant #DCR-860615

Rights and permissions

Reprints and permissions

About this article

Cite this article

Clote, P., Hájek, P. & Paris, J. On some formalized conservation results in arithmetic. Arch Math Logic 30, 201–218 (1990). https://doi.org/10.1007/BF01792983

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01792983

Keywords

Navigation