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On the Broader Epistemological Significance of Self-Justifying Axiom Systems

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Logic, Language, Information, and Computation (WoLLIC 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8652))

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Abstract

This article will be a continuation of our research into self-justifying systems. It will introduce several new theorems (one of which will transform our previous infinite-sized self-verifying logics into formalisms or purely finite size). It will explain how self-justification is useful, even when the Incompleteness Theorem clearly limits its scope.

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References

  1. Adamowicz, Z.: Herbrand consistency and bounded arithmetic. Fundamenta Mathematicae 171(3), 279–292 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  2. Adamowicz, Z., Zbierski, P.: On Herbrand consistency in weak theories. Archive for Mathematical Logic 40(6), 399–413 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bezboruah, A., Shepherdson, J.C.: Gödel’s second incompleteness theorem for Q. Jour. Symb. Logic 41(2), 503–512 (1976)

    MATH  MathSciNet  Google Scholar 

  4. Buss, S.R.: Bounded Arithmetic. Studies in Proof Theory, Lecture Notes 3. Published by Bibliopolis (1986) (Revised version of Ph. D. Thesis)

    Google Scholar 

  5. Buss, S.R., Ignjatovic, A.: Unprovability of consistency statements in fragments of bounded arithmetic. Annals Pure and Applied Logic 74(3), 221–244 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  6. Dawson, J.W.: Logical Dilemmas the life and work of Kurt Gödel. AKPeters (1997)

    Google Scholar 

  7. Feferman, S.: Arithmetization of metamathematics in a general setting. Fundamenta Mathematicae 49, 35–92 (1960)

    MATH  MathSciNet  Google Scholar 

  8. Fitting, M.: First Order Logic and Automated Theorem Proving. Springer (1996)

    Google Scholar 

  9. Friedman, H.M.: On the consistency, completeness and correctness problems. Technical report, Ohio State Univ. (1979), See Pudlák [28]’s summary of this result

    Google Scholar 

  10. Friedman, H.M.: Translatability and relative consistency. Technical report, Ohio State Univ. (1979), See Pudlák [28]’s summary of this result

    Google Scholar 

  11. Gödel, K.: Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatshefte für Math. Phys. 38, 173–198 (1931)

    Article  Google Scholar 

  12. Gödel, K.: The present situation in the foundations of mathematics. In: Feferman, S., Dawson, J.W., Goldfarb, W., Parson, C., Solovay, R. (eds.) Collected Works Volume III: Unpublished Essays and Lectures, pp. 45–53. Oxford University Press (2004), Our quotes from this 1933 lecture come from its page 52

    Google Scholar 

  13. Goldstein, R.: Incompleteness The Proof and Paradox of Kurt Gödel. Norton (2005)

    Google Scholar 

  14. Hájek, P.: Mathematical fuzzy logic and natural numbers. Fundamenta Mathematicae 81, 155–163 (2007)

    MATH  Google Scholar 

  15. Hájek, P.: Towards metamathematics of weak arithmetics over fuzzy. Logic Journal of the IPL 19(3), 467–475 (2011)

    Article  MATH  Google Scholar 

  16. Hájek, P., Pudlák, P.: Metamathematics of First Order Arithmetic. Springer (1991)

    Google Scholar 

  17. Hilbert, D.: Über das Unendliche. Mathematische Annalen 95, 161–191 (1926)

    Article  MathSciNet  Google Scholar 

  18. Hilbert, D., Bernays, P.: Grundlagen der Mathematik, vol. II. Springer (1939)

    Google Scholar 

  19. Jeroslow, R.G.: Consistency statements in formal theories. Fundamenta Mathematicae 72, 17–40 (1971)

    MATH  MathSciNet  Google Scholar 

  20. Kleene, S.C.: On notation for ordinal numbers. Jour. Symb. Logic 3(1), 150–156 (1938)

    Article  MathSciNet  Google Scholar 

  21. Kołodziejczyk, L.A.: On the Herbrand notion of consistency for finitely axiomatizable fragments of bounded arithmetic theories. Jour. Symb. Logic 71(2), 624–638 (2006)

    Article  MATH  Google Scholar 

  22. Kreisel, G., Takeuti, G.: Formally self-referential propositions in cut-free classical analysis and related systems. Dissertationes Mathematicae 118, 1–50 (1974)

    MathSciNet  Google Scholar 

  23. Löb, M.H.: A solution of a problem of Leon Henkin. Jour. Symb. Logic 20(2), 115–118 (1955)

    Article  MATH  Google Scholar 

  24. Nelson, E.: Predicative Arithmetic. Math. Notes. Princeton Univ. Press (1986)

    Google Scholar 

  25. Parikh, R.: Existence and feasibility in arithmetic. Jour. Symb. Logic 36(3), 494–508 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  26. Parsons, C.H.: On n −quantifier induction. Jour. Symb. Logic 37(3), 466–482 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  27. Pudlák, P.: Cuts, consistency statements and interpretations. Jour. Symb. Logic 50(2), 423–442 (1985)

    Article  MATH  Google Scholar 

  28. Pudlák, P.: On the lengths of proofs of consistency. In: Collegium Logicum: Annals of the Kurt Gödel, vol. 2, pp. 65–86. Springer-Wien (1996)

    Google Scholar 

  29. Rogers, H.A.: Theory of Recursive Functions and Effective Compatibility. McGraw Hill (1967)

    Google Scholar 

  30. Salehi, S.: Herbrand consistency of some arithmetical theories. Jour. Symb. Logic 77(3), 807–827 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  31. Solovay, R.M.: Telephone conversation in 1994 describing Solovay’s generalization of one of Pudlák’s theorems [27], using some methods of Nelson and Wilkie-Paris [24, 34] (The Appendix A of [37] offers a 4-page summary of this conversation) (1994)

    Google Scholar 

  32. Svejdar, V.: An interpretation of Robinson arithmetic in its Grzegorczjk’s weaker variant. Fundamenta Mathematicae 81, 347–354 (2007)

    MATH  MathSciNet  Google Scholar 

  33. Visser, A.: Faith and falsity. Annals Pure and Applied Logic 131(1), 103–131 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  34. Wilkie, A.J., Paris, J.B.: On the scheme of induction for bounded arithmetic. Annals Pure and Applied Logic 35, 261–302 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  35. Willard, D.E.: Self-verifying axiom systems. In: Mundici, D., Gottlob, G., Leitsch, A. (eds.) KGC 1993. LNCS, vol. 713, pp. 325–336. Springer, Heidelberg (1993)

    Chapter  Google Scholar 

  36. Willard, D.E.: The semantic tableaux version of the second incompleteness theorem extends almost to Robinson’s arithmetic Q. In: Dyckhoff, R. (ed.) TABLEAUX 2000. LNCS, vol. 1847, pp. 415–430. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  37. Willard, D.E.: Self-verifying systems, the incompleteness theorem and the tangibility reflection principle. Jour. Symb. Logic 66(2), 536–596 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  38. Willard, D.E.: How to extend the semantic tableaux and cut-free versions of the second incompleteness theorem almost to Robinson’s arithmetic Q. Jour. Symb. Logic 67(1), 465–496 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  39. Willard, D.E.: A version of the second incompleteness theorem for axiom systems that recognize addition but not multiplication as a total function. In: Hendricks, V., Neuhaus, F., Pederson, S.A., Scheffler, U., Wansing, H. (eds.) First Order Logic Revisited, pp. 337–368. Logos Verlag, Berlin (2004)

    Google Scholar 

  40. Willard, D.E.: An exploration of the partial respects in which an axiom system recognizing solely addition as a total function can verify its own consistency. Jour. Symb. Logic 70(4), 1171–1209 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  41. Willard, D.E.: A generalization of the second incompleteness theorem and some exceptions to it. Annals Pure and Applied Logic 141(3), 472–496 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  42. Willard, D.E.: On the available partial respects in which an axiomatization for real valued arithmetic can recognize its consistency. Jour. Symb. Logic 71(4), 1189–1199 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  43. Willard, D.E.: Passive induction and a solution to a Paris-Wilkie open question. Annals Pure and Applied Logic 146(2), 124–149 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  44. Willard, D.E.: Some specially formulated axiomizations for IΣ0 manage to evade the Herbrandized version of the second incompleteness theorem. Information and Computation 207(10), 1078–1093 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  45. Yourgrau, P.: A World Without Time: The Forgotten Legacy of Gödel and Einstein. Basic Books (2005), See page 58 for the passages we have quoted

    Google Scholar 

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Authors and Affiliations

  1. University at Albany Computer Science and Mathematics Departments, USA

    Dan E. Willard

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  1. Dan E. Willard
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Editors and Affiliations

  1. Fachbereich Mathematik, TU Darmstadt, Schlossgartenstrasse 7, 64289, Darmstadt, Germany

    Ulrich Kohlenbach

  2. Department of Computer Science, Universidad de Chile, Avenida Blanco Encalada 2120, Oficina 316, Tercer Piso, Santiago, Chile

    Pablo Barceló

  3. Centro de Informática, Universidade Federal de Pernambuco (UFPE), Av. Jornalista Anibal Fernandes, s/n, Cidade Universitária, 50., 740-560, Recife, PE, Brazil

    Ruy de Queiroz

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Willard, D.E. (2014). On the Broader Epistemological Significance of Self-Justifying Axiom Systems. In: Kohlenbach, U., Barceló, P., de Queiroz, R. (eds) Logic, Language, Information, and Computation. WoLLIC 2014. Lecture Notes in Computer Science, vol 8652. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44145-9_16

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  • DOI: https://doi.org/10.1007/978-3-662-44145-9_16

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-44144-2

  • Online ISBN: 978-3-662-44145-9

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