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This paper identifies the fundamental error inherent in Gödel’s proof of his Incompleteness Theorem. The error is generated by the ambiguity of the language of Gödel’s outline proof of his Proposition V, a proposition for which Gödel declined to furnish a detailed proof. The error arises from a confusion of the meta-language and the languages to which it refers, a confusion which is exacerbated b y the failure of Gödel to clarify the principal assertions involved in his suggested proof outline, where there is a reliance on intuition rather than logical transparency.
The result of this vagueness of presentation is that there is no clear delineation of the meta-language and the sub-languages involved, with the result that an equivalence is asserted between an expression of the meta - language and a sub - language, an equivalence which is logically untenable. It is shown here that this means that the self - reference generated by Proposition VI in Gödel’s proof relies on this erroneous intuitive assumption, and hence the self-reference of that Proposition is logically untenable, and there is no logical basis for Gödel’s result.
This paper uses straightforward logic and does not rely on any philosophical or semantical arguments.