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

  • Dan E. Willard (a1)
  • DOI: http://dx.doi.org/10.2307/2695030
  • Published online: 01 March 2014
Abstract
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.

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[10]M. Fitting , First order logic and automated theorem proving, Monographs in Computer Science, Springer-Verlag, 1990.

[17]J. Krajícek , A note on the proofs of falsehoods, Archive for Mathematical Logic, vol. 26 (1987), pp. 169176.

[18]J. Krajícek , Bounded propositional logic and complexity theory, Cambridge University Press, 1995.

[20]G. Kreisel , 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.

[23]E. Mendelson , Introduction to mathematical logic, Mathematics Series, Wadsworth and Brooks/Cole, 1987.

[24]E. Nelson , Predicative arithmetic, Mathematical Notes, Princeton University Press, 1986.

[29]P. Pudlák , 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.

[33]R. Smullyan , First order logic, Springer-Verlag, 1968.

[39]A. Wilkie and J. Paris , On the scheme of induction for bounded arithmetic, Annals of Pure Applied Logic, vol. 35 (1987), pp. 261302.

[44]D. Willard , 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]C. Wrathall , Rudimentary predicates and relative computation, SIAM Journal on Computing, vol. 7 (1978), pp. 194209.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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