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).

**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.

© 1996-9

1999-05-25