Hostname: page-component-8448b6f56d-sxzjt Total loading time: 0 Render date: 2024-04-24T04:22:48.593Z Has data issue: false hasContentIssue false
  • English
  • Français

Self-verifying axiom systems, the incompleteness theorem and related reflection principles

Published online by Cambridge University Press:  12 March 2014

Dan E. Willard
Dan E. Willard*
Affiliation:
Department of Computer Science, State University of New York, Albany, New York 12222, USA, E-mail: dew@cs.albany.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We will study several weak axiom systems that use the Subtraction and Division primitives (rather than Addition and Multiplication) to formally encode the theorems of Arithmetic. Provided such axiom systems do not recognize Multiplication as a total function, we will show that it is feasible for them to verify their Semantic Tableaux, Herbrand, and Cut-Free consistencies. If our axiom systems additionally do not recognize Addition as a total function, they will be capable of recognizing the consistency of their Hilbert-style deductive proofs. Our axiom systems will not be strong enough to recognize their Canonical Reflection principle, but they will be capable of recognizing an approximation of it, called the “Tangibility Reflection Principle”. We will also prove some new versions of the Second Incompleteness Theorem stating essentially that it is not possible to extend our exceptions to the Incompleteness Theorem much further.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 66 , Issue 2 , June 2001 , pp. 536 - 596
Copyright
Copyright © Association for Symbolic Logic 2001

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Adamowicz, Z., On tableau consistency in weak theories, circulating manuscript from the Mathematics Institute of the Polish Academy of Sciences, 1999.Google Scholar
[2]Benett, J., Ph. D. dissertation, Princeton University, 1962.Google Scholar
[3]Bezboruah, A. and Shepherdson, J., Gödel’s second incompleteness theorem for Q, this Journal, vol. 41 (1976), pp. 503–512.Google Scholar
[4]Boolos, G., The logic of provability, Cambridge University Press, 1993.Google Scholar
[5]Buss, S., Bounded arithmetic, Ph. d. dissertation, Princeton University, published in Proof Theory Lecture Notes, vol. 3, Bibliopolic. 1986.Google Scholar
[6]Buss, S., Polynomial hierarchy and fragments of bounded arithmetic, 17th ACM Symposium on theory of computation, 1985, pp. 285–290.Google Scholar
[7]Enderton, H., A mathematical introduction to logic, Academic Press, 1972.Google Scholar
[8]Feferman, S., Arithmetization of metamathematics in a general setting, Fundamenta Mathematicae, vol. 49 (1960), pp. 35–92.CrossRefGoogle Scholar
[9]Feferman, S., Transfinite recursive progressions of axiomatic theories, this Journal, vol. 27 (1962), pp. 259–316.Google Scholar
[10]Fitting, M., First order logic and automated theorem proving, Monographs in Computer Science, Springer-Verlag, 1990.CrossRefGoogle Scholar
[11]Gentzen, G., Collected papers of Gerhard Gentzen, North-Holland, 1969, English translation by Szabo, M..Google Scholar
[12]Hájek, P., On the interpretatability of theories containing arithmetic (II), Communications in Mathematics from the University of Carolina, vol. 22 (1981), pp. 595–594.Google Scholar
[13]Hájek, P. and Pudlák, P., Metamathematics of first order arithmetic, Springer-Verlag, 1991.Google Scholar
[14]Hilbert, D. and Bernays, B., Grundlagen der Mathematik, vol. 1 (1934) and vol. 2 (1939), Springer-Verlag.Google Scholar
[15]Jeroslow, R., Consistency statements in formal mathematics, Fundamenta Mathematicae, vol. 72 (1971), pp. 17–40.CrossRefGoogle Scholar
[16]Kleene, S., On the notation of ordinal numbers, this Journal, vol. 3 (1938), pp. 150–156.Google Scholar
[17]Krajícek, J., A note on the proofs of falsehoods, Archive for Mathematical Logic, vol. 26 (1987), pp. 169–176.CrossRefGoogle Scholar
[18]Krajícek, J., Bounded propositional logic and complexity theory, Cambridge University Press, 1995.CrossRefGoogle Scholar
[19]Krajícek, J. and Pudlák, P., Propositional proof systems, the consistency of first-order theories and the complexity of computation, this Journal, vol. 54 (1989), pp. 1063–1079.Google Scholar
[20]Kreisel, G., A survey of proof theory, Part I, Journal of Symbolic Logic, vol. (1968), pp. 321–388 and Part II, in Proceedings of second Scandinavian logic symposium (with Fenstad, editor), North-Holland Press, Amsterdam, 1971.CrossRefGoogle Scholar
[21]Kreisel, G. and Takeuti, G., Formally self-referential propositions for cut-free classical analysis and related systems, Dissertations Mathematica, vol. 118 (1974), pp. 1–55.Google Scholar
[22]Löb, M., A solution to a problem by Leon Henkin, this Journal, vol. 20 (1955). pp. 115–118.Google Scholar
[23]Mendelson, E., Introduction to mathematical logic, Mathematics Series, Wadsworth and Brooks/Cole, 1987.CrossRefGoogle Scholar
[24]Nelson, E., Predicative arithmetic, Mathematical Notes, Princeton University Press, 1986.CrossRefGoogle Scholar
[25]Parikh, R., Existence and feasibility in arithmetic, this Journal, vol. 36 (1971), pp. 494–508.Google Scholar
[26]Paris, J. and Dimitracopoulos, C., A note on the undefinability of cuts, this Journal, vol. 48 (1983), pp. 564–569.Google Scholar
[27]Paris, J. and Wilkie, A., Δ0 sets and induction, Proceedings of the Jadswin logic conference (Poland), Leeds University Press, 1981, pp. 237–248.Google Scholar
[28]Pudlák, P., Cuts consistency statements and interpretations, this Journal, vol. 50 (1985), pp. 423–442.Google Scholar
[29]Pudlák, P., On the lengths of proofs of consistency, Collegium logicum: Annals of the Kurt Gödel Society, vol. 2, Springer-Verlag, Wien and New York, 1996, published in cooperation with the Kurt Gödel Gesellshaft of the Institut für Computersprachen of Technische Universität Wien (Vienna. Austria), pp. 65–86.CrossRefGoogle Scholar
[30]Rogers, H., Theory of recursive functions and effective compatibility, McGraw-Hill, 1967.Google Scholar
[31]Shoenfeld, J., Mathematical logic, Addison-Wesley, 1967.Google Scholar
[32]Smorynski, C., The incompleteness theorem, Handbook on mathematical logic, 1983, pp. 821–866.Google Scholar
[33]Smullyan, R., First order logic, Springer-Verlag, 1968.CrossRefGoogle Scholar
[34]Solovay, R., Private communications, generalizing Pudlák’s Proposition 2.2 from [28] to establish that no axiom system can recognize Successor, Subtraction and Division as functions, prove all the Π1 theorems of Arithmetic and verify its own Hilbert consistency. (Solovay never published his theorem, and he gave us permission to present a short summary of his proof in Appendix A. Solovay’s full theorem is slightly stronger than the result proven in Appendix A.), 04 1994.Google Scholar
[35]Statman, R., Herbrand’s theorem and Gentzen’s notion of a direct proof, Handbook on mathematical logic, North-Holland Publishing House, 1983, pp. 897–913.Google Scholar
[36]Takeuti, G., On a generalized logical calculus, Japan Journal on Mathematics, vol. 23 (1953), pp. 39–96.Google Scholar
[37]Takeuti, G., Proof theory, Studies in Logic, vol. 81, North-Holland, 1987.Google Scholar
[38]Wilkie, A., an unpublished manuscript which indicates that there exists a cut of Robinson’s System Q that models IΣ0 + Ωn in a global rather than local manner. (The localized version of the same theorem was published by Nelson in [24].) Hájek and Pudlák [13] credit Wilkie’s unpublished theorem (on page 407) for being responsible for Theorem 5.7 in Chapter V.5.c of their textbook, and they give a formal statement and proof of Wilkie’s unpublished theorem.Google Scholar
[39]Wilkie, A. and Paris, J., On the scheme of induction for bounded arithmetic, Annals of Pure Applied Logic, vol. 35 (1987), pp. 261–302.CrossRefGoogle Scholar
[40]Willard, D., Self-verifying axiom systems, Proceedings of the third Kurt Gödel symposium, 1993, published in Lecture Notes in Computer Science, vol. 173, Springer-Verlag, See also the next reference., pp. 325–336.Google Scholar
[41]Willard, D., Self-verifying axiom systems and the incompleteness theorem, Technical report, Suny-Albany, 03 1994, this 50-page technical report expands the abbreviated proofs in the 12-page Extended Abstract [40] into full proofs.Google Scholar
[42]Willard, D., Self-reflection principles and NP-hardness, Dimacs Series in Discrete Mathematics and Theoretical Computer Science, vol. 39, American Mathematics Society, 12 1997.Google Scholar
[43]Willard, D., The tangibility reflection principle for self-verifying axiom systems, The proceedings of the third Kurt Gödel colloquium, Lecture Notes in Computer Science, vol. 1289, Springer-verlag, 1997, pp. 319–334.Google Scholar
[44]Willard, D., The Semantic Tableaux version of the Second Incompleteness Theorem extends almost to Robinson’s Arithmetic Q, Automated reasoning with Semantic Tableaux and related methods, no. 1847, Springer-Verlag, 2000, pp. 415–430.CrossRefGoogle Scholar
[45]Wrathall, C., Rudimentary predicates and relative computation, SIAM Journal on Computing, vol. 7 (1978), pp. 194–209.CrossRefGoogle Scholar