Chapter

Gödel Numbering and Gödel’s Theorem

J. R. LUCAS

in The Freedom of the Will

Published in print September 1970 | ISBN: 9780198243434
Published online October 2011 | e-ISBN: 9780191680687 | DOI: http://dx.doi.org/10.1093/acprof:oso/9780198243434.003.0024
Gödel Numbering and Gödel’s Theorem

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Gödel devised a method whereby the formulae of a given formal language could be put in a 1-1 correspondence with a subset of the natural numbers: that is to say, to every number there is correlated only one formula, and to every formula there is correlated one and only one number, which we call its Gödel number. In order to pursue the possibility of reproducing the Liar paradox within formal mathematics we need to follow up Gödel's scheme for correlating numbers and formulae with a method of making up properties of numbers and relations between numbers that will correspond to logically important properties of formulae and relations between formulae. But it turns out that this is not possible. Truth — and similarly falsity — are not representable in the logistic calculus L, though provability-in-the-logistic-calculus L — the next best thing — is. In Gödel's theorem, therefore, the Liar paradox is altered by substituting ‘unprovable-in-the-logistic-calculus-L’ for ‘untrue’.

Keywords: Gödel number; formal language; natural numbers; Liar paradox; logistic calculus

Chapter.  2131 words. 

Subjects: Metaphysics

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