Consistency, Turing computability and gödel's first incompleteness theorem
Minds and Machines 18 (1) (2008)
| Abstract | It is well understood and appreciated that Gödel’s Incompleteness Theorems apply to sufficiently strong, formal deductive systems. In particular, the theorems apply to systems which are adequate for conventional number theory. Less well known is that there exist algorithms which can be applied to such a system to generate a gödel-sentence for that system. Although the generation of a sentence is not equivalent to proving its truth, the present paper argues that the existence of these algorithms, when conjoined with Gödel’s results and accepted theorems of recursion theory, does provide the basis for an apparent paradox. The difficulty arises when such an algorithm is embedded within a computer program of sufficient arithmetic power. The required computer program (an AI system) is described herein, and the paradox is derived. A solution to the paradox is proposed, which, it is argued, illuminates the truth status of axioms in formal models of programs and Turing machines. | |||||||||
| Keywords | No keywords specified (fix it) | |||||||||
| Categories | ||||||||||
| Options |
|
|||||||||
Download options| PhilPapers Archive |
Upload a copy of this paper Check publisher's policy on self-archival Papers currently archived: 5,865 |
| External links |
 |
| Through your library | Configure |
Similar books and articlesCarlo Cellucci (1992). Gödel's Incompleteness Theorem and the Philosophy of Open Systems. In Daniel Miéville (ed.), Kurt Gödel: Actes du Colloque, Neuchâtel 13-14 Juin 1991, pp. 103-127. Travaux de logique N. 7, Université de Neuchâtel.
Richard Tieszen (1994). Mathematical Realism and Gödel's Incompleteness Theorems. Philosophia Mathematica 2 (3):177-201.
Francesco Berto (2009). There's Something About Gödel: The Complete Guide to the Incompleteness Theorem. Wiley-Blackwell.
Raymond M. Smullyan (1992). Gödel's Incompleteness Theorems. Oxford University Press.
Roman Murawski (1997). Gödel's Incompleteness Theorems and Computer Science. Foundations of Science 2 (1):123-135.
George Boolos (2007). Computability and Logic. Cambridge University Press.
Francesco Berto (2009). The Gödel Paradox and Wittgenstein's Reasons. Philosophia Mathematica 17 (2):208-219.
Carlo Cellucci (1993). From Closed to Open Systems. In J. Czermak (ed.), Philosophy of Mathematics, pp. 206-220. Hölder-Pichler-Tempsky.
Panu Raatikainen (2005). On the Philosophical Relevance of Gödel's Incompleteness Theorems. Revue Internationale de Philosophie 59 (4):513-534.
Analytics
Monthly downloads |
Added to index2009-01-28Total downloads40 ( #29,458 of 556,807 )Recent downloads (6 months)1 ( #64,847 of 556,807 )How can I increase my downloads? |
My notes
Discussion
