Takeuti's conjecture

In mathematics, Takeuti's conjecture is the conjecture of Gaisi Takeuti that a sequent formalisation of second-order logic has cut-elimination (Takeuti 1953). It was settled positively:

  • By Tait, using a semantic technique for proving cut-elimination, based on work by Schütte (Tait 1966);
  • Independently by Takahashi by a similar technique (Takahashi 1967);
  • It is a corollary of Jean-Yves Girard's syntactic proof of strong normalization for System F.

Takeuti's conjecture is equivalent to the consistency of second-order arithmetic in the sense that each of the statements can be derived from each other in the weak system PRA; consistency refers here to the truth of the Gödel sentence for second-order arithmetic. It is also equivalent to the strong normalization of the Girard/Reynold's System F.

See also

References

  • William W. Tait, 1966. A nonconstructive proof of Gentzen's Hauptsatz for second order predicate logic. In Bulletin of the American Mathematical Society, 72:980983.
  • Gaisi Takeuti, 1953. On a generalized logic calculus. In Japanese Journal of Mathematics, 23:3996. An errata to this article was published in the same journal, 24:149156, 1954.
  • Moto-o Takahashi, 1967. A proof of cut-elimination in simple type theory. In Japanese Mathematical Society, 10:4445.


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