How to extend the semantic tableaux and cut-free versions of the second incompleteness theorem almost to Robinson's arithmetic Q
David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Ezio Di Nucci
Jack Alan Reynolds
Learn more about PhilPapers
Journal of Symbolic Logic 67 (1):465-496 (2002)
Let us recall that Raphael Robinson's Arithmetic Q is an axiom system that differs from Peano Arithmetic essentially by containing no Induction axioms , . We will generalize the semantic-tableaux version of the Second Incompleteness Theorem almost to the level of System Q. We will prove that there exists a single rather long Π 1 sentence, valid in the standard model of the Natural Numbers and denoted as V, such that if α is any finite consistent extension of Q + V then α will be unable to prove its Semantic Tableaux consistency. The same result will also apply to axiom systems α with infinite cardinality when these infinite-sized axiom systems satisfy a minor additional constraint, called the Conventional Encoding Property. Our formalism will also imply that the semantic-tableaux version of the Second Incompleteness Theorem generalizes for the axiom system IΣ 0 , as well as for all its natural extensions. (This answers an open question raised twenty years ago by Paris and Wilkie .)
|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.
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.
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.
Similar books and articles
Carl Mummert & Stephen G. Simpson (2004). An Incompleteness Theorem for [Image]. Journal of Symbolic Logic 69 (2):612 - 616.
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.
Panu Raatikainen (2005). On the Philosophical Relevance of Gödel's Incompleteness Theorems. Revue Internationale de Philosophie 59 (4):513-534.
Bernhard Beckert & Rajeev GorÉ (2001). Free-Variable Tableaux for Propositional Modal Logics. Studia Logica 69 (1):59-96.
Carlo Cellucci (2000). Analytic Cut Trees. Logic Journal of the IGPL 8 (6):733-750.
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 (2001). Self-Verifying Axiom Systems, the Incompleteness Theorem and Related Reflection Principles. Journal of Symbolic Logic 66 (2):536-596.
Added to index2009-01-28
Total downloads8 ( #404,558 of 1,911,306 )
Recent downloads (6 months)4 ( #178,269 of 1,911,306 )
How can I increase my downloads?