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Effective bounds from ineffective proofs in analysis: An application of functional interpretation and majorization

Published online by Cambridge University Press:  12 March 2014

Ulrich Kohlenbach
Ulrich Kohlenbach*
Affiliation:
Fachbereich Mathematik, J.-W.-Goethe-Universität, W 6000 Frankfurt-Am-Main, Germany
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Abstract

We show how to extract effective bounds Φ for ⋀u1⋀v ≤ytu⋁wηG0-sentences which depend on u only (i.e. ⋀u⋀v ≤y, tu⋁w ≤η ΦuG0) from arithmetical proofs which use analytical assumptions of the form

(ϒ, δ, ρ, and τ are arbitrary finite types, η ≤ 2, G0 and F0 are quantifier-free, and s and t are closed terms). If τ ≤ 2, (*) can be weakened to

This is used to establish new conservation results about weak Konig's lemma. Applications to proofs in classical analysis, especially uniqueness proofs in approximation theory, will be given in subsequent papers.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 57 , Issue 4 , December 1992 , pp. 1239 - 1273
Copyright
Copyright © Association for Symbolic Logic 1992

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References

REFERENCES

Bezem, M. A. [1985], Strongly majorizable functionals of finite type: a model for bar recursion containing discontinuous functionals, this Journal, vol. 50, pp. 652–660.Google Scholar
Bridges, D. [1980], A constructive development of Chebyshev approximation theory, Journal of Approximation Theory, vol. 30, pp. 99–120.CrossRefGoogle Scholar
Bridges, D. [1982], Lipschitz constants and moduli of continuity for the Chebyshev projection, Proceedings of the American Mathematical Society, vol. 85, pp. 557–561.CrossRefGoogle Scholar
Clote, P., Hájek, P., and Paris, J. [1990], On some formalized conservation results in arithmetic, Archive for Mathematical Logic, vol. 30, pp. 201–218.CrossRefGoogle Scholar
Poussin, Ch.-J. de la Vallée [1919], Leçons sur l'approximation des fonctions d'une variable réelle, Gauthier-Villars, Paris.Google Scholar
Feferman, S. [1977], Theories of finite type related to mathematical practice, Handbook of mathematical logic (Barwise, J., editor), North-Holland, Amsterdam, 1977, pp. 913–971.CrossRefGoogle Scholar
Feferman, S. [1988], Hubert's program relativized: proof-theoretical and foundational reductions, this Journal, vol. 53, pp. 364–384.Google Scholar
Friedman, H. [1975], Some systems of second order arithmetic and their use, Proceedings of the international congress of mathematicians (Vancouver, 1974), Vol. 1, Canadian Mathematical Congress, Montreal, pp. 235–242.Google Scholar
Friedman, H. [1976], Systems of second order arithmetic with restricted induction (abstract), this Journal, vol. 41, pp. 558–559.Google Scholar
Gödel, K. [1933], Zur intuitionistischen Arithmetik und Zahlentheorie, Ergebnisse eines Mathematischen Kolloquiums, vol. 4, pp. 34–38.Google Scholar
Gödel, K. [1958], Über eine bisher noch nicht benütze Erweiterung des finiten Standpunktes, Dialectica, vol. 12, pp. 280–287.CrossRefGoogle Scholar
Hilbert, D. [1926], Über das Unendliche, Mathematische Amalen, vol. 95, pp. 161–190.CrossRefGoogle Scholar
Howard, W. A. [1973], Hereditarily majorizable functionals of finite type, in Troelstra [1973], pp. 454–461.Google Scholar
Kirby, L. and Paris, J. B. [1977], Initial segments of models of Peano's axioms, Set theory and hierarchy theory, V (Lachlan, A.et al., editors), Lecture Notes in Mathematics, vol. 619, Springer-Verlag, Berlin, pp. 211–226.CrossRefGoogle Scholar
Kleene, S. C. [1952], Introduction to metamathematics, Van Nostrand, Princeton, New Jersey.Google Scholar
Kleene, S. C. [1959], Recursive functionals and quantifiers of finite type. I, Transactions of the American Mathematical Society, vol. 91, pp. 1–52.Google Scholar
Kleene, S. C. and Vesley, R. E. [1965], The foundations of intuitionistic mathematics, especially in relation to recursive functions, North-Holland, Amsterdam.Google Scholar
Kohlenbach, U. [1990], Theorie der majorisierbaren und stetigen Funktionale und ihre Anwendung bei der Extraktion von Schranken aus inkonstruktiven Beweisen: Effektive Eindeutigkeitsmodule bei besten Approximationen aus ineffektiven Eindeutigkeitsbeweisen, Dissertation, Frankfurt.Google Scholar
Kohlenbach, U. [1992], Pointwise hereditary majorization and some applications, Archive for Mathematical Logic, vol. 31, pp. 227–241.CrossRefGoogle Scholar
Kohlenbach, U. [1992A], Remarks on Herhrand normal forms and Herbrand realizations, Archive for Mathematical Logic, vol. 31, pp. 305–317.CrossRefGoogle Scholar
Kreisel, G. [1963], Preface, Seminar on the foundations of analysis. Reports, Vol. II, Stanford University, Stanford, California.Google Scholar
Kreisel, G. [1966], Review of Kleene and Vesley [1965], this Journal, vol. 31, pp. 258–261.Google Scholar
Kreisel, G., Mints, G. E., and Simpson, S. G. [1975], The use of abstract language in elementary metamathematics: some pedagogic examples, Logic colloquium, Boston 1972–1973 (Parikh, R., editor), Lecture Notes in Mathematics, vol. 453, Springer-Verlag, Berlin, pp. 38–131.Google Scholar
Luckhardt, H. [1973], Extensional Gödei functional interpretation: a consistency proof of classical analysis, Lecture Notes in Mathematics, vol. 306, Springer-Verlag, Berlin.CrossRefGoogle Scholar
Natanson, I. P. [1949], Konstruktive Funktionentheorie, GITTL, Moscow (Russian); German translation, Akademie-Verlag, Berlin, 1955.Google Scholar
Scott, D. S. [1962], Algebras of sets binumerable in complete extensions of arithmetic, Recursive function theory (Dekker, J. C. E., editor), Proceedings of Symposia in Pure Mathematics, vol. 5, American Mathematical Society, Providence, Rhode Island, pp. 117–121.CrossRefGoogle Scholar
Shoenfield, J. R. [1954], A relative consistency proof, this Journal, vol. 19, pp. 21–28.Google Scholar
Sieg, W. [1985], Fragments of arithmetic, Annals of Pure and Applied Logic, vol. 28, pp. 33–71.CrossRefGoogle Scholar
Sieg, W. [1987], Konig's lemma and fragments of arithmetic, this Journal, vol. 52, pp. 342–343 (abstract).Google Scholar
Sieg, W. [1987A], Provably recursive functionals of theories with König's lemma, Rendiconti del Seminario Matematico, Università e Politecnico di Torino, Special issue: Conference on logic and computer science: new trends and issues, pp. 75–92.Google Scholar
Sieg, W. [1991], Herbrand analyses, Archive for Mathematical Logic, vol. 30, pp. 409–441.CrossRefGoogle Scholar
Tait, W. W. [1965], Infinitely long terms of transfinite type, Formal systems and recursive functions (Crossley, J. and Dummett, M., editors), North-Holland, Amsterdam, pp. 176–185.CrossRefGoogle Scholar
Troelstra, A. S. (editor) [1973], Metamathematical investigation of intuitionistic arithmetic and analysis, Lecture Notes in Mathematics, vol. 344, Springer-Verlag, Berlin.CrossRefGoogle Scholar
Troelstra, A. S. [1974], Note on the fan theorem, this Journal, vol. 39, pp. 584–596.Google Scholar
Troelstra, A. S. [1977], Some models for intuitionistic finite type arithmetic with fan functional, this Journal, vol. 42, pp. 194–202.Google Scholar