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The semantic version of the proof The semantic version of the proof
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Three noteworthy features of the proof Three noteworthy features of the proof
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The non-semantic version of the proof The non-semantic version of the proof
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5 (page 63)p. 63C5The diagonal proof of Gödel’s theorem
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Published:November 2022
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Abstract
This chapter presents what is called the ‘diagonal’ proof of Gödel’s theorem. Two versions of the proof are given: a semantic version which proceeds on the assumption that any statement in any given formal language is either true or false; and a non-semantic version which dispenses with that assumption. Each proof begins by in effect assuming—contrary to what is to be proved—that there is a theory that is sufficiently strong, consistent, complete, and axiomatizable. An argument is given that, if there is such a theory, then there is an algorithm for telling whether any given statement belongs to it; and that, if there is such an algorithm, then, via a Gödel numbering, there is a set of natural numbers that is picked out by some expression in the language of the theory but that cannot be picked out by any expression in the language of the theory on pain of contradiction. Three features of the proof are highlighted: the connection that it establishes between axiomatizability and the existence of such an algorithm; the power that it shows that any such algorithm would have if it did exist; and the generous conception of sufficient strength on which it rests. The third of these raises a concern, namely that the conception is overgenerous. An assurance is given that this concern can be assuaged.
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