Arithmetic Proof and Open Sentences
David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Ezio Di Nucci
Jack Alan Reynolds
Learn more about PhilPapers
Philosophy Study 2 (1):43-50 (2012)
If the concept of proof (including arithmetic proof) is syntactically restricted to closed sentences (or their Gödel numbers), then the standard accounts of Gödel’s Incompleteness Theorems (and Löb’s Theorem) are blocked. In these standard accounts (Gödel’s own paper and the exposition in Boolos’ Computability and Logic are treated as exemplars), it is assumed that certain formulas (notably so called “Gödel sentences”) containing the Gödel number of an open sentence and an arithmetic proof predicate are closed sentences. Ordinary usage of the term “provable” (and indeed “unprovable”) favors their restriction to closed sentences which unlike so-called open sentences can be true or false. In this paper the restricted form of provability is called strong provability or unprovability. If this concept of proof is adopted, then there is no obvious alternative path to establishing those theorems.
|Keywords||Logic Godel's Thorem Proof Theory|
|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
Samuel R. Buss (1994). On Gödel's Theorems on Lengths of Proofs I: Number of Lines and Speedup for Arithmetics. Journal of Symbolic Logic 59 (3):737-756.
Gregor Damschen (2011). Questioning Gödel's Ontological Proof: Is Truth Positive? European Journal for Philosophy of Religion 3 (1):161-169.
Peter Milne (2007). On Gödel Sentences and What They Say. Philosophia Mathematica 15 (2):193-226.
Stewart Shapiro (2002). Incompleteness and Inconsistency. Mind 111 (444):817-832.
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.
Hirohiko Kushida (2010). The Modal Logic of Gödel Sentences. Journal of Philosophical Logic 39 (5):577 - 590.
Zofia Adamowicz & Teresa Bigorajska (2001). Existentially Closed Structures and Gödel's Second Incompleteness Theorem. Journal of Symbolic Logic 66 (1):349-356.
Andrzej Mostowski (1952/1982). Sentences Undecidable in Formalized Arithmetic: An Exposition of the Theory of Kurt Gödel. Greenwood Press.
Markus Pantsar (2009). Truth, Proof and Gödelian Arguments: A Defence of Tarskian Truth in Mathematics. Dissertation, University of Helsinki
Added to index2012-05-29
Total downloads5 ( #505,915 of 1,796,539 )
Recent downloads (6 months)2 ( #346,486 of 1,796,539 )
How can I increase my downloads?