The Kreisel conjecture is a conjecture in proof theory that postulates that, if there is a uniform bound to the lengths of shortest proofs of instances of , then the universal generalization is necessarily provable in Peano arithmetic. A special case of the conjecture was proven true by M. Baaz in 1988 (Baaz and Pudlák 1993).

Baaz, M. and Pudlák P. "Kreisel's Conjecture for . In Arithmetic, Proof Theory, and Computational Complexity, Papers from the Conference Held in Prague, July 2-5, 1991 (Ed. P. Clote and J. Krajiček). New York: Oxford University Press, pp. 30-60, 1993.

Dawson, J. "The Gödel Incompleteness Theorem from a Length of Proof Perspective." Amer. Math. Monthly 86, 740-747, 1979.

Kreisel, G. "On the Interpretation of Nonfinitistic Proofs, II." J. Symbolic Logic 17, 43-58, 1952.

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Weisstein, Eric W. "Kreisel Conjecture." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/KreiselConjecture.html