A type of mathematical result which is considered by most logicians as more natural than the metamathematical incompleteness results first discovered by Gödel. Finite combinatorial examples include Goodstein's theorem, a finite form of Ramsey's theorem, and a finite form of Kruskal's tree theorem (Kirby and Paris 1982; Smorynski 1980, 1982, 1983; Gallier 1991).
Natural Independence Phenomenon
See alsoGödel's First Incompleteness Theorem, Gödel's Second Incompleteness Theorem, Goodstein's Theorem, Kruskal's Tree Theorem, Ramsey's Theorem
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ReferencesGallier, J. "What's so Special about Kruskal's Theorem and the Ordinal Gamma? A Survey of Some Results in Proof Theory." Ann. Pure and Appl. Logic 53, 199-260, 1991.Kirby, L. and Paris, J. "Accessible Independence Results for Peano Arithmetic." Bull. London Math. Soc. 14, 285-293, 1982.Smorynski, C. "Some Rapidly Growing Functions." Math. Intell. 2, 149-154, 1980.Smorynski, C. "The Varieties of Arboreal Experience." Math. Intell. 4, 182-188, 1982.Smorynski, C. "'Big' News from Archimedes to Friedman." Not. Amer. Math. Soc. 30, 251-256, 1983.
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Weisstein, Eric W. "Natural Independence Phenomenon." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/NaturalIndependencePhenomenon.html