An exploration of the partial respects in which an axiom system recognizing solely addition as a total function can verify its own consistency
David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Ezio Di Nucci
Jonathan Jenkins Ichikawa
Jack Alan Reynolds
Learn more about PhilPapers
Journal of Symbolic Logic 70 (4):1171-1209 (2005)
This article will study a class of deduction systems that allow for a limited use of the modus ponens method of deduction. We will show that it is possible to devise axiom systems α that can recognize their consistency under a deduction system D provided that: (1) α treats multiplication as a 3-way relation (rather than as a total function), and that (2) D does not allow for the use of a modus ponens methodology above essentially the levels of Π1 and Σ1 formulae. Part of what will make this boundary-case exception to the Second Incompleteness Theorem interesting is that we will also characterize generalizations of the Second Incompleteness Theorem that take force when we only slightly weaken the assumptions of our boundary-case exceptions in any of several further directions
|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
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.
Zofia Adamowicz & Paweł Zbierski (2001). On Herbrand Consistency in Weak Arithmetic. Archive for Mathematical Logic 40 (6):399-413.
C. Dimitracopoulos (1989). Overspill and Fragments of Arithmetic. Archive for Mathematical Logic 28 (3):173-179.
Samuel R. Buss & Aleksandar Ignjatović (1995). Unprovability of Consistency Statements in Fragments of Bounded Arithmetic. Annals of Pure and Applied Logic 74 (3):221-244.
Z. Ratajczyk (1993). Subsystems of True Arithmetic and Hierarchies of Functions. Annals of Pure and Applied Logic 64 (2):95-152.
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.
Walter Carnielli (2011). The Single-Minded Pursuit of Consistency and its Weakness. Studia Logica 97 (1):81 - 100.
Similar books and articles
Dan E. Willard (2001). Self-Verifying Axiom Systems, the Incompleteness Theorem and Related Reflection Principles. Journal of Symbolic Logic 66 (2):536-596.
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.
Fang-Wen Yuan (2008). “The Strict Deduction System Is Impossible to Derive the Contradiction” And the Proof. Proceedings of the Xxii World Congress of Philosophy 13:147-162.
T. Thacher Robinson (1968). Independence of Two Nice Sets of Axioms for the Propositional Calculus. Journal of Symbolic Logic 33 (2):265-270.
Raymond D. Gumb (2001). An Extended Joint Consistency Theorem for a Nonconstructive Logic of Partial Terms with Definite Descriptions. Studia Logica 69 (2):279-292.
A. Avron (1998). Multiplicative Conjunction and an Algebraic Meaning of Contraction and Weakening. Journal of Symbolic Logic 63 (3):831-859.
Sachio Hirokawa (1996). The Proofs of Α→Α in P - W. Journal of Symbolic Logic 61 (1):195-211.
Sachio Hirokawa, Yuichi Komori & Misao Nagayama (2000). A Lambda Proof of the P-W Theorem. Journal of Symbolic Logic 65 (4):1841-1849.
Torben BraÜner (2005). Natural Deduction for First-Order Hybrid Logic. Journal of Logic, Language and Information 14 (2):173-198.
Added to index2010-08-24
Total downloads13 ( #284,877 of 1,934,581 )
Recent downloads (6 months)1 ( #434,193 of 1,934,581 )
How can I increase my downloads?