Skip to main content

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

  • Dan E. Willard (a1)

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.

Hide All
[1]Adamowicz, Z., On tableau consistency in weak theories, circulating manuscript from the Mathematics Institute of the Polish Academy of Sciences, 1999.
[2]Benett, J., Ph. D. dissertation, Princeton University, 1962.
[3]Bezboruah, A. and Shepherdson, J., Gödel’s second incompleteness theorem for Q, this Journal, vol. 41 (1976), pp. 503512.
[4]Boolos, G., The logic of provability, Cambridge University Press, 1993.
[5]Buss, S., Bounded arithmetic, Ph. d. dissertation, Princeton University, published in Proof Theory Lecture Notes, vol. 3, Bibliopolic. 1986.
[6]Buss, S., Polynomial hierarchy and fragments of bounded arithmetic, 17th ACM Symposium on theory of computation, 1985, pp. 285290.
[7]Enderton, H., A mathematical introduction to logic, Academic Press, 1972.
[8]Feferman, S., Arithmetization of metamathematics in a general setting, Fundamenta Mathematicae, vol. 49 (1960), pp. 3592.
[9]Feferman, S., Transfinite recursive progressions of axiomatic theories, this Journal, vol. 27 (1962), pp. 259316.
[10]Fitting, M., First order logic and automated theorem proving, Monographs in Computer Science, Springer-Verlag, 1990.
[11]Gentzen, G., Collected papers of Gerhard Gentzen, North-Holland, 1969, English translation by Szabo, M..
[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.
[13]Hájek, P. and Pudlák, P., Metamathematics of first order arithmetic, Springer-Verlag, 1991.
[14]Hilbert, D. and Bernays, B., Grundlagen der Mathematik, vol. 1 (1934) and vol. 2 (1939), Springer-Verlag.
[15]Jeroslow, R., Consistency statements in formal mathematics, Fundamenta Mathematicae, vol. 72 (1971), pp. 1740.
[16]Kleene, S., On the notation of ordinal numbers, this Journal, vol. 3 (1938), pp. 150156.
[17]Krajícek, J., A note on the proofs of falsehoods, Archive for Mathematical Logic, vol. 26 (1987), pp. 169176.
[18]Krajícek, J., Bounded propositional logic and complexity theory, Cambridge University Press, 1995.
[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. 10631079.
[20]Kreisel, G., A survey of proof theory, Part I, Journal of Symbolic Logic, vol. (1968), pp. 321388 and Part II, in Proceedings of second Scandinavian logic symposium (with Fenstad, editor), North-Holland Press, Amsterdam, 1971.
[21]Kreisel, G. and Takeuti, G., Formally self-referential propositions for cut-free classical analysis and related systems, Dissertations Mathematica, vol. 118 (1974), pp. 155.
[22]Löb, M., A solution to a problem by Leon Henkin, this Journal, vol. 20 (1955). pp. 115118.
[23]Mendelson, E., Introduction to mathematical logic, Mathematics Series, Wadsworth and Brooks/Cole, 1987.
[24]Nelson, E., Predicative arithmetic, Mathematical Notes, Princeton University Press, 1986.
[25]Parikh, R., Existence and feasibility in arithmetic, this Journal, vol. 36 (1971), pp. 494508.
[26]Paris, J. and Dimitracopoulos, C., A note on the undefinability of cuts, this Journal, vol. 48 (1983), pp. 564569.
[27]Paris, J. and Wilkie, A., Δ0 sets and induction, Proceedings of the Jadswin logic conference (Poland), Leeds University Press, 1981, pp. 237248.
[28]Pudlák, P., Cuts consistency statements and interpretations, this Journal, vol. 50 (1985), pp. 423442.
[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. 6586.
[30]Rogers, H., Theory of recursive functions and effective compatibility, McGraw-Hill, 1967.
[31]Shoenfeld, J., Mathematical logic, Addison-Wesley, 1967.
[32]Smorynski, C., The incompleteness theorem, Handbook on mathematical logic, 1983, pp. 821866.
[33]Smullyan, R., First order logic, Springer-Verlag, 1968.
[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.
[35]Statman, R., Herbrand’s theorem and Gentzen’s notion of a direct proof, Handbook on mathematical logic, North-Holland Publishing House, 1983, pp. 897913.
[36]Takeuti, G., On a generalized logical calculus, Japan Journal on Mathematics, vol. 23 (1953), pp. 3996.
[37]Takeuti, G., Proof theory, Studies in Logic, vol. 81, North-Holland, 1987.
[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.
[39]Wilkie, A. and Paris, J., On the scheme of induction for bounded arithmetic, Annals of Pure Applied Logic, vol. 35 (1987), pp. 261302.
[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. 325336.
[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.
[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.
[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. 319334.
[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. 415430.
[45]Wrathall, C., Rudimentary predicates and relative computation, SIAM Journal on Computing, vol. 7 (1978), pp. 194209.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 11 *
Loading metrics...

Abstract views

Total abstract views: 310 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 26th September 2018. This data will be updated every 24 hours.