Journal of Symbolic Logic

An exploration of the partial respects in which an axiom system recognizing solely addition as a total function can verify its own consistency

Dan E. Willard

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

This article will study a class of deduction systems that allow for a limited use of the modus ponens method of deduction. We will show that it is possible to devise axiom systems α that can recognize their consistency under a deduction system D provided that: (1) α treats multiplication as a 3-way relation (rather than as a total function), and that (2) D does not allow for the use of a modus ponens methodology above essentially the levels of Π1 and Σ1 formulae.

Part of what will make this boundary-case exception to the Second Incompleteness Theorem interesting is that we will also characterize generalizations of the Second Incompleteness Theorem that take force when we only slightly weaken the assumptions of our boundary-case exceptions in any of several further directions.

Article information

Source
J. Symbolic Logic Volume 70, Issue 4 (2005), 1171-1209.

Dates
First available: 18 October 2005

Permanent link to this document
http://projecteuclid.org/euclid.jsl/1129642122

Digital Object Identifier
doi:10.2178/jsl/1129642122

Mathematical Reviews number (MathSciNet)
MR2194244

Zentralblatt MATH identifier
1102.03055

Citation

Willard, Dan E. An exploration of the partial respects in which an axiom system recognizing solely addition as a total function can verify its own consistency. Journal of Symbolic Logic 70 (2005), no. 4, 1171--1209. doi:10.2178/jsl/1129642122. http://projecteuclid.org/euclid.jsl/1129642122.


Export citation

References

  • Z. Adamowicz Herbrand consistency and bounded arithmetic, Fundamenta Mathematicae, vol. 171 (2002), pp. 279--292.
  • Z. Adamowicz and T. Bigorajska Existentially closed structures and Gödel's second incompleteness theorem, Journal of Symbolic Logic, vol. 66 (2001), pp. 349--356.
  • Z. Adamowicz and P. Zbierski The logic of mathematics: A modern course in classical logic, John Wiley and Sons,1997.
  • T. Arai Derivability conditions on Rosser's proof predicates, Notre Dame Journal of Formal Logic, vol. 31 (1990), pp. 487--497.
  • J. Benett Ph.D. thesis, Princeton University,1962, A detailed summary of Benett's main theorem can be found on pages 299--303 and 406 of the Hájek-Pudlák textbook [?].
  • A. Bezboruah and J. C. Shepherdson Gödel's second incompleteness theorem for Q, Journal of Symbolic Logic, vol. 41 (1976), pp. 503--512.
  • S. R. Buss Bounded arithmetic, Proof Theory Lecture Notes, no. 3, Bibliopolis,1986, (Revised version of Ph. D. Thesis.).
  • S. R. Buss and A. Ignjatovic Unprovability of consistency statements in fragments of bounded arithmetic, Annals of Pure and Applied Logic, vol. 74 (1995), pp. 221--244.
  • C. Dimitracopoulos Overspill and fragments of arithmetic, Archive for Mathematical Logic, vol. 28 (1989), pp. 173--179.
  • S. Feferman Arithmetization of mathematics in a general setting, Fundamenta Mathematicae, vol. 19 (1960), pp. 35--92.
  • S. Feferman, G. Kriesel, and S. Orey 1-consistency and faithful interpretations, Archive for Mathematical Logic, vol. 6 (1962), pp. 52--63.
  • M. Fitting First order logic and automated theorem proving, Springer,1990.
  • H. M. Friedman On the consistency, completeness and correctness problems, Technical report, Ohio State University Mathematics Department,1979, Some summaries of these unpublished results by Friedman can be found in [?].
  • K. Gödel Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, Monatshefte für Mathematik und Physik, vol. 37 (1931), pp. 349--360.
  • P. Hájek On interpretability in set theory Part I, Commentationes Mathematicae Universitatis Carolinae, vol. 12 (1971), pp. 73--79.
  • P. Hájek and P. Pudlák Metamathematics of first order arithmetic, Springer,1991.
  • D. Hilbert and P. Bernays Grundlagen der Mathematik, Springer,1939.
  • R. G. Jeroslow Consistency statements in formal mathematics, Fundamenta Mathematicae, vol. 72 (1971), pp. 17--40.
  • R. Kaye Models of Peano Arithmetic, Oxford University Press,1991.
  • S. C. Kleene On the notation of ordinal numbers, Journal of Symbolic Logic, vol. 3 (1938), pp. 150--156.
  • J. Krajícek Bounded propositional logic and complexity theory, Cambridge University Press,1995.
  • G. Kreisel and G. Takeuti Formally self-referential propositions for cut-free classical analysis, Dissertationes Mathematicae, vol. 118 (1974), pp. 1--55.
  • P. Lindström On faithful interpretability, Computation and proof theory, Lecture Notes in Mathematics, vol. 1104, Springer,1984, pp. 279--288.
  • M. H. Löb A solution to a problem by Leon Henkin, Journal of Symbolic Logic, vol. 20 (1955), pp. 115--118.
  • E. Mendelson Introduction to mathematical logic, Chapman Hall,1997.
  • R. Montague Theories incomparable with respect to interpretability, Journal of Symbolic Logic, vol. 27 (1962), pp. 195--211.
  • J. Mycieslski A lattice connected with relative interpretability, Notices of the American Mathematical Society, vol. 9 (1962), pp. 407--408.
  • E. Nelson Predicative arithmetic, Princeton Math Notes Press,1986.
  • S. Orey Relative interpretations, Zeitschrift für Mathematische Logik, vol. 7 (1961), pp. 146--153.
  • R. Parikh Existence and feasibility in arithmetic, Journal of Symbolic Logic, vol. 36 (1971), pp. 494--508.
  • J. B. Paris and C. Dimitracopoulos A note on the undefinability of cuts, Journal of Symbolic Logic, vol. 48 (1983), pp. 564--569.
  • J. B. Paris and A. J. Wilkie $\Delta_0$ sets and induction, Proceedings of the Jadswin logic conference, Leeds University Press,1981, pp. 237--248.
  • P. Pudlák Some prime elements in the lattice of interpretability, Transactions of the American Mathematical Society, vol. 280 (1983), pp. 255--275.
  • Z. Ratajczyk Subsystems of true arithmetic and hierarchies of functions, Annals of Pure and Applied Logic, vol. 64 (1993), pp. 95--152.
  • H. A. Rogers Recursive functions and effective compatibility, McGraw Hill,1967.
  • J. B. Rosser Extensions of some earlier theorems by Gödel and Church, Journal of Symbolic Logic, vol. 1 (1936), pp. 87--91.
  • S. Salehi Herbrand consistency in arithmetics with bounded induction, Ph.D. thesis, Polish Academy,2001.
  • C. A. Smoryński The incompleteness theorem, Handbook on mathematical logic (J. Barwise, editor), North Holland,1977, pp. 821--865.
  • R. M. Smullyan First order logic, Springer,1968.
  • R. M. Solovay Injecting inconsistencies into models of PA, Annals of Pure and Applied Logic, vol. 44 (1988), pp. 102--132.
  • V. Svejdar Degrees of interpretability, Commentationes Mathematicae Universitatis Carolinae, vol. 19 (1978), pp. 783--813.
  • G. Takeuti Proof theory, North Holland,1987.
  • A. Tarski, A. Mostowski, and R. Robinson Undecidable theories, North Holland Press,1953.
  • A. Visser Interpretability logic, Mathematical Logic: Proceedings of the Heyting Summer School,1988, pp. 175--208.
  • P. Vop\v enka and P. Hájek Existence of a generalized semantic model of Gödel-Bernays set theory, Bulletin de l'Académie Polonaise des Sciences, Mathématiques, Astronomiques et Physiques, vol. 12 (1973), pp. 1079--1086.
  • A. J. Wilkie and J. B. Paris On the scheme of induction for bounded arithmetic, Annals of Pure and Applied Logic, vol. 35 (1987), pp. 261--302.
  • D. Willard Self-verifying axiom systems, Proceedings of the third kurt gödel's symposium, Lecture Notes in Computer Science, vol. 713,1993, pp. 325--336.
  • C. Wrathall Rudimentary predicates and relative computation, SIAM Journal on Computing, vol. 7 (1978), pp. 194--209.