Skip to main content Accessibility help
×
Home
  • Print publication year: 2010
  • Online publication date: March 2011

Proof interpretations and majorizability

Summary

Abstract. In the last fifteen years, the traditional proof interpretations of modified realizability and functional (dialectica) interpretation in finite-type arithmetic have been adapted by taking into account majorizability considerations. One of such adaptations, the monotone functional interpretation of Ulrich Kohlenbach, has been at the center of a vigorous program in applied proof theory dubbed proof mining. We discuss some of the traditional and majorizability interpretations, including the recent bounded interpretations, and focus on the main theoretical techniques behind proof mining.

Introduction. Functional interpretations were introduced half a century ago by Kurt Gödel in. Göodel's interpretation uses functionals of finite type and is an exact interpretation. It is exact in the sense that it provides precise witnesses of existential statements. Another example of an exact (functional) interpretation is Georg Kreisel's modified realizability. In the last fifteen years or so, there has been an interest in interpretations which are not exact, but only demand bounds for existential witnesses. These interpretations, when dealing with bounds for functionals of every type, are based on the majorizability notions of William Howard and Marc Bezem. This is the case with Ulrich Kohlenbach's monotone modified realizability and monotone functional interpretation, the bounded modified realizability of Fernando Ferreira and Ana Nunes, or the bounded functional interpretation of Ferreira and Paulo Oliva. There are, to be sure, other interpretations which also incorporate majorizability notions to a certain extent: for instance, the seminal Diller-Nahm interpretation, Wolfgang Burr's interpretation of KPω or the very recent interpretation of Jeremy Avigad and Henry Towsner which is able to provide a proof theoretic analysis of ID1.

Related content

Powered by UNSILO
REFERENCES
[1] Y., Akama, S., Berardi, S., Hayashi, and U., Kohlenbach, An arithmetical hierarchy of the law of excluded middle and related principles, Proceedings of the 19th Annual IEEE Symposium on Logic and Computer Science, vol. 117, IEEE Press, 2004, pp. 192–201.
[2] J., Avigad and S., Feferman, Gödel's functional (“Dialectica”) interpretation, Handbook of Proof Theory (S. R., Buss, editor), Studies in Logic and the Foundations of Mathematics, vol. 137, North Holland, Amsterdam, 1998, pp. 337–405.
[3] J., Avigad and H., Towsner, Functional interpretation and inductive definition, The Journal of Symbolic Logic, vol. 74 (2009), pp. 1100–1120.
[4] M., Bezem, Strongly majorizable functionals of finite type: a model for bar recursion containing discontinuous functionals, The Journal of Symbolic Logic, vol. 50 (1985), pp. 652–660.
[5] W., Burr, A Diller-Nahm-style functional interpretation of KPω, Archive for Mathematical Logic, vol. 39 (2000), pp. 599–604.
[6] J., Diller and W., Nahm, Eine Variante zur Dialectica-Interpretation der Heyting-Arithmetik endlicher Typen, Archive für mathematische Logik und Grundlagenforschung, vol. 16 (1974), pp. 49–66.
[7] P., Engrácia and F., Ferreira, The bounded functional interpretation of the double negation shift, to appear in The Journal of Symbolic Logic.
[8] F., Ferreira, A most artistic package of a jumble of ideas, dialectica, vol. 62 (2008), pp. 205–222, Special Issue: Gödel's dialectica Interpretation. Guest editor: Thomas, Strahm.
[9] F., Ferreira, On a new functional interpretation (abstract), The Bulletin of Symbolic Logic, vol. 14 (2008), pp. 128–129.
[10] F., Ferreira, Injecting uniformities into Peano arithmetic, Annals of Pure and Applied Logic, vol. 157 (2009), pp. 122–129, Special Issue: Kurt Gödel Centenary Research Prize Fellowships. Editors: Sergei, Artemov, Matthias, Baaz and Harvey, Friedman.
[11] F., Ferreira and A., Nunes, Bounded modified realizability, The Journal of Symbolic Logic, vol. 71 (2006), pp. 329–346.
[12] F., Ferreira and P., Oliva, Bounded functional interpretation, Annals of Pure and Applied Logic, vol. 135 (2005), pp. 73–112.
[13] F., Ferreira, Bounded functional interpretation and feasible analysis, Annals of Pure and Applied Logic, vol. 145 (2005), pp. 115–129.
[14] J., Gaspar, Factorization of the Shoenfield-like bounded functional interpretation, Notre Dame Journal of Formal Logic, vol. 50 (2009), pp. 53–60.
[15] P., Gerhardy and U., Kohlenbach, General logical metatheorems for functional analysis, Transactions of the American Mathematical Society, vol. 360 (2008), pp. 2615–2660.
[16] K., Gödel, Zur intuitionistischen Arithmetik und Zahlentheorie, Ergebnisse eines Mathematischen Kolloquiums, vol. 4 (1933), pp. 34–38.
[17] K., Gödel, Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes, dialectica, vol. 12 (1958), pp. 280–287, Reprinted with an English translation in [19].
[18] K., Gödel, On an extension of finitary mathematics which has not yet been used, 1972, Published in [19], pages 271–280. Revised version of [17].
[19] K., Gödel, Collected Works, vol. II, (S., Feferman, editor), Oxford University Press, Oxford, 1990.
[20] W. A., Howard, Hereditarily majorizable functionals of finite type, Metamathematical investigation of intuitionistic Arithmetic and Analysis (A. S., Troelstra, editor), Lecture Notes in Mathematics, vol. 344, Springer, Berlin, 1973, pp. 454–461.
[21] S. C., Kleene, On the interpretation of intuitionistic number theory, The Journal of Symbolic Logic, vol. 10 (1945), pp. 109–124.
[22] S. C., Kleene, Countable functionals, Constructivity in Mathematics (A., Heyting, editor), North Holland, Amsterdam, 1959, pp. 81–100.
[23] U., Kohlenbach, Theorie der majorisierbaren und stetigen Funktionale und ihre Anwendung bei der Extraktion von Schranken aus inkonstruktiven Beweisen: Effektive Eindeutigkeitsmodule bei besten Approximationen aus ineffektiven Beweisen, Ph.D. thesis, Frankfurt, pp. xxii+278, 1990.
[24] U., Kohlenbach, Effective bounds from ineffective proofs in analysis: an application of functional interpretation and majorization, The Journal of Symbolic Logic, vol. 57 (1992), pp. 1239–1273.
[25] U., Kohlenbach, Pointwise hereditary majorization and some applications, Archive for Mathematical Logic, vol. 31 (1992), pp. 227–241.
[26] U., Kohlenbach, Effective moduli from ineffective uniqueness proofs. An unwinding of de La Vallée Poussin's proof for Chebycheff approximation, Annals of Pure and Applied Logic, vol. 64 (1993), pp. 27–94.
[27] U., Kohlenbach, Analysing proofs in analysis, Logic: from Foundations to Applications (W., Hodges, M., Hyland, C., Steinhorn, and J., Truss, editors), Oxford University Press, 1996, pp. 225–260.
[28] U., Kohlenbach, Mathematically strong subsystems of analysis with low rate of growth of provably recursive functionals, Archive for Mathematical Logic, vol. 36 (1996), pp. 31–71.
[29] U., Kohlenbach, Relative constructivity, The Journal of Symbolic Logic, vol. 63 (1998), pp. 1218–1238.
[30] U., Kohlenbach, On the no-counterexample interpretation, The Journal of Symbolic Logic, vol. 64 (1999), pp. 1491–1511.
[31] U., Kohlenbach, Intuitionistic choice and restricted classical logic, Mathematical Logic Quarterly, vol. 47 (2001), pp. 455–460.
[32] U., Kohlenbach, Foundational and mathematical uses of higher types, Reflections on the Foundations of Mathematics: Essay in Honor of Solomon Feferman (W., Sieg et al., editors), Lecture Notes in Logic, vol. 15, A. K. Peters, 2002, pp. 92–116.
[33] U., Kohlenbach, Some logical metatheorems with applications in functional analysis, Transactions of the American Mathematical Society, vol. 357 (2005), pp. 89–128.
[34] U., Kohlenbach, Proof interpretations and the computational content of proofs in mathematics, Bulletin of the EATCS, vol. 93 (2007), pp. 143–173.
[35] U., Kohlenbach, Applied Proof Theory: Proof Interpretations and Their Use in Mathematics, Springer Monographs in Mathematics, Springer, Berlin, 2008.
[36] U., Kohlenbach, Effective bounds from proofs in abstract functional analysis, New Computational Paradigms: Changing Conceptions of What is Computable (S. B., Cooper, B., Löwe, and A., Sorbi, editors), Springer Verlag, to appear.
[37] U., Kohlenbach and P., Oliva, Proof mining: a systematic way of analysing proofs in mathematics, Proceedings of the Steklov Institute of Mathematics, vol. 242 (2003), pp. 136–164.
[38] U., Kohlenbach and T., Streicher, Shoenfield is Gödel after Krivine, Mathematical Logic Quarterly, vol. 53 (2007), pp. 176–179.
[39] G., Kreisel, On the interpretation of non-finitist proofs, part I, The Journal of Symbolic Logic, vol. 16 (1951), pp. 241–267.
[40] G., Kreisel, Interpretation of analysis by means of constructive functionals of finite types, Constructivity in Mathematics (A., Heyting, editor), North Holland, Amsterdam, 1959, pp. 101–128.
[41] G., Kreisel, On weak completeness of intuitionistic predicate logic, The Journal of Symbolic Logic, vol. 27 (1962), pp. 139–158.
[42] H., Luckhardt, Extensional Gödel Functional Interpretation: A Consistency Proof of Classical Analysis, Lecture Notes in Mathematics, vol. 306, Springer, Berlin, 1973.
[43] P., Oliva, Unifying functional interpretations, Notre Dame Journal of Formal Logic, vol. 47 (2006), pp. 263–290.
[44] C., Parsons, On a number theoretic choice schema and its relation to induction, Intuitionism and Proof Theory (J., Myhill, A., Kino, and R. E., Vesley, editors), North-Holland, 1970, pp. 459–473.
[45] J. R., Shoenfield, Mathematical Logic, Addison-Wesley Publishing Company, 1967, Republished in 2001 by AK Peters.
[46] S. G., Simpson, Subsystems of Second Order Arithmetic, Perspectives in Mathematical Logic, Springer, Berlin, 1999.
[47] C., Spector, Provably recursive functionals of analysis: a consistency proof of analysis by an extension of principles in current intuitionistic mathematics, Recursive Function Theory: Proceedings of Symposia in Pure Mathematics (F. D. E., Dekker, editor), vol. 5, AMS, Providence, Rhode Island, 1962, pp. 1–27.
[48] T., Strahm (editor), Gödel's dialectica Interpretation, dialectica, vol. 62 (2008), pp. 145–290.
[49] W., Tait, Intentional interpretations of functionals of finite type I, The Journal of Symbolic Logic, vol. 32 (1967), pp. 198–212.
[50] T., Tao, The correspondence principle and finitary ergodic theory, Essay posted August 30, 2008. Available at: http://terrytao.wordpress.com/2008/08/30/the-correspondence-principle-and-finitary-ergodic-theory/.
[51] T., Tao, Soft analysis, hard analysis, and the finite convergence principle, Structure and Randomness: Pages from Year One of a Mathematical Blog, AMS, 2008, pp. 17–29.
[52] A. S., Troelstra (editor), Metamathematical Investigation of Intuitionistic Arithmetic and Analysis, Lecture Notes in Mathematics, vol. 344, Springer, Berlin, 1973.
[53] A. S., Troelstra, Introductory note to [17] and [18], 1990, Published in [19], pages 214–241.
[54] A. S., Troelstra, Realizability, Handbook of Proof Theory (S. R., Buss, editor), Studies in Logic and the Foundations of Mathematics, vol. 137, North Holland, Amsterdam, 1998, pp. 408–473.
[55] A. S., Troelstra and D., van Dalen, Constructivism in Mathematics. An Introduction, Studies in Logic and the Foundations of Mathematics, vol. 121, North Holland, Amsterdam, 1988.
[56] M., Yasugi, Intuitionistic analysis and Gödel's interpretation, Journal of the Mathematical Society of Japan, vol. 15 (1963), pp. 101–112.