David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Ezio Di Nucci
Jack Alan Reynolds
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Journal of Symbolic Logic 66 (2):536-596 (2001)
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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References found in this work BETA
A. J. Wilkie & J. B. Paris (1987). On the Scheme of Induction for Bounded Arithmetic Formulas. Annals of Pure and Applied Logic 35 (3):261-302.
Citations of this work BETA
Dan E. Willard (2006). A Generalization of the Second Incompleteness Theorem and Some Exceptions to It. Annals of Pure and Applied Logic 141 (3):472-496.
Dan E. Willard (2007). Passive Induction and a Solution to a Paris–Wilkie Open Question. Annals of Pure and Applied Logic 146 (2):124-149.
Greg Hjorth (2002). 2001-2002 Winter Meeting of the Association for Symbolic Logic. Bulletin of Symbolic Logic 8 (2):312-318.
Stan S. Wainer (2006). 2005 Summer Meeting of the Association for Symbolic Logic. Logic Colloquium'05. Bulletin of Symbolic Logic 12 (2):310-361.
John Steel (2006). 2005 Annual Meeting of the Association for Symbolic Logic. Bulletin of Symbolic Logic 12 (1):143-167.
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