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 also

Gödel's First Incompleteness Theorem, Gödel's Second Incompleteness Theorem, Goodstein's Theorem, Kruskal's Tree Theorem, Ramsey's Theorem## Explore with Wolfram|Alpha

## 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.

## Referenced on Wolfram|Alpha

Natural Independence Phenomenon## Cite this as:

Weisstein, Eric W. "Natural Independence Phenomenon."
From *MathWorld*--A Wolfram Web Resource. https://mathworld.wolfram.com/NaturalIndependencePhenomenon.html