David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
Learn more about PhilPapers
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
|Keywords||No keywords specified (fix it)|
|Categories||categorize this paper)|
Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)
|Through your library|
References found in this work BETA
No references found.
Citations of this work BETA
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.
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.
Similar books and articles
Robert E. Beaudoin (1987). Strong Analogues of Martin's Axiom Imply Axiom R. Journal of Symbolic Logic 52 (1):216-218.
Carlo Cellucci (1993). From Closed to Open Systems. In J. Czermak (ed.), Philosophy of Mathematics, pp. 206-220. Hölder-Pichler-Tempsky.
David W. Miller (2007). Some Restricted Lindenbaum Theorems Equivalent to the Axiom of Choice. Logica Universalis 1 (1):183-199.
Dan E. Willard (2006). On the Available Partial Respects in Which an Axiomatization for Real Valued Arithmetic Can Recognize its Consistency. Journal of Symbolic Logic 71 (4):1189-1199.
Dan E. Willard (2002). How to Extend the Semantic Tableaux and Cut-Free Versions of the Second Incompleteness Theorem Almost to Robinson's Arithmetic Q. Journal of Symbolic Logic 67 (1):465-496.
Dan E. Willard (2005). 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 (4):1171-1209.
Added to index2009-01-28
Total downloads7 ( #192,321 of 1,099,789 )
Recent downloads (6 months)1 ( #303,541 of 1,099,789 )
How can I increase my downloads?