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This paper develops fuzzy and neutrosophic extensions of Gödel's incompleteness theorems based on graded and triadic self-reference. Starting from fuzzy and neutrosophic forms of the Liar Paradox — originally formulated by Florentin Smarandache — we construct generalized Gödel sentences through diagonalization in formal systems capable of encoding their own provability. In the fuzzy setting, incompleteness appears as nontrivial intermediate provability degrees: the Gödel sentence stabilizes at a truth degree F ∈ (0,1) rather than collapsing to a binary undecidable. In the Neutrosophic setting, incompleteness appears as an irreducible valuation ⟨T, I, F⟩ with simultaneous nonzero truth, indeterminacy, and falsehood components.
This paper develops fuzzy and neutrosophic extensions of Gödel's incompleteness theorems based on graded and triadic self-reference. Starting from fuzzy and neutrosophic forms of the Liar Paradox — originally formulated by Florentin Smarandache — we construct generalized Gödel sentences through d...
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