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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).
See also Gödel's Incompleteness Theorem, Goodstein's Theorem, Kruskal's Tree Theorem, Ramsey's Theorem
References
Gallier, J. ``What's so Special about Kruskal's Theorem and the Ordinal Gamma[0]?
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.