Can Gödel's Incompleteness Theorem be a Ground for Dialetheism?

Korean Journal of Logic 20 (2):241-271 (2017)
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Abstract
Dialetheism is the view that there exists a true contradiction. This paper ventures to suggest that Priest’s argument for Dialetheism from Gödel’s theorem is unconvincing as the lesson of Gödel’s proof (or Rosser’s proof) is that any sufficiently strong theories of arithmetic cannot be both complete and consistent. In addition, a contradiction is derivable in Priest’s inconsistent and complete arithmetic. An alternative argument for Dialetheism is given by applying Gödel sentence to the inconsistent and complete theory of arithmetic. We argue, however, that the alternative argument raises a circularity problem. In sum, Gödel’s and its related theorem merely show the relation between a complete and a consistent theory. A contradiction derived by the application of Gödel sentence has the value of true sentences, i.e. the both-value, only under the inconsistent models for arithmetic. Without having the assumption of inconsistency or completeness, a true contradiction is not derivable from the application of Gödel sentence. Hence, Gödel’s and its related theorem never can be a ground for Dialetheism.
Keywords Gödel’s incompleteness theorem  Rosser’s incompleteness theorem  Dialetheism  Inconsistent arithmetic
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