David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
Learn more about PhilPapers
Mind 111 (444):817-832 (2002)
Graham Priest's In Contradiction (Dordrecht: Martinus Nijhoff Publishers, 1987, chapter 3) contains an argument concerning the intuitive, or ‘naïve’ notion of (arithmetic) proof, or provability. He argues that the intuitively provable arithmetic sentences constitute a recursively enumerable set, which has a Gödel sentence which is itself intuitively provable. The incompleteness theorem does not apply, since the set of provable arithmetic sentences is not consistent. The purpose of this article is to sharpen Priest's argument, avoiding reference to informal notions, consensus, or Church's thesis. We add Priest's dialetheic semantics to ordinary Peano arithmetic PA, to produce a recursively axiomatized formal system PA★ that contains its own truth predicate. Whether one is a dialetheist or not, PA★ is a legitimate, rigorously defined formal system, and one can explore its proof‐theoretic properties. The system is inconsistent (but presumably non‐trivial), and it proves its own Gödel sentence as well as its own soundness. Although this much is perhaps welcome to the dialetheist, it has some untoward consequences. There are purely arithmetic (indeed, Π0) sentences that are both provable and refutable in PA★. So if the dialetheist maintains that PA★ is sound, then he must hold that there are true contradictions in the most elementary language of arithmetic. Moreover, the thorough dialetheist must hold that there is a number g which both is and is not the code of a derivation of the indicated Gödel sentence of PA★. For the thorough dialetheist, it follows ordinary PA and even Robinson arithmetic are themselves inconsistent theories. I argue that this is a bitter pill for the dialetheist to swallow
|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
No citations found.
Similar books and articles
Aldo Ursini (1978). On the Set of 'Meaningful' Sentences of Arithmetic. Studia Logica 37 (3):237 - 241.
Laureano Luna & Alex Blum (2008). Arithmetic and Logic Incompleteness: The Link. The Reasoner 2 (3):6.
Raymond M. Smullyan (1985). Uniform Self-Reference. Studia Logica 44 (4):439 - 445.
Francesco Berto (2009). The Gödel Paradox and Wittgenstein's Reasons. Philosophia Mathematica 17 (2):208-219.
Philip Hugly & Charles Sayward (1989). Can There Be a Proof That an Unprovable Sentence of Arithmetic is True? Dialectica 43 (43):289-292.
Francesco Berto (2009). There's Something About Gödel: The Complete Guide to the Incompleteness Theorem. Wiley-Blackwell.
John R. Lucas (1961). Minds, Machines and Godel. Philosophy 36 (April-July):112-127.
Robert F. Hadley (2008). Consistency, Turing Computability and Gödel's First Incompleteness Theorem. Minds and Machines 18 (1):1-15.
Panu Raatikainen (2005). On the Philosophical Relevance of Gödel's Incompleteness Theorems. Revue Internationale de Philosophie 59 (4):513-534.
Added to index2009-01-28
Total downloads57 ( #42,570 of 1,699,706 )
Recent downloads (6 months)5 ( #128,702 of 1,699,706 )
How can I increase my downloads?