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Parity and the Pigeonhole Principle

  • M. Ajtai
Part of the Progress in Computer Science and Applied Logic book series (PCS, volume 9)

Abstract

The Pigeonhole Principle is the statement that there is no one-to-one map of a set of size n into a set of size n — 1. This is a theorem of Peano Arithmetic that is it can be proved using the axioms of complete induction. A weaker version of Peano Arithmetic is I Δ0 where we allow only bounded formulas in the induction axioms, that is for each bounded formula \( \phi (\vec x,z)\) the corresponding induction axiom \( \forall \vec x\left( {\left( {\varphi \left( {\vec x,0} \right) \wedge \forall \left( {\varphi \left( {\vec x,y} \right) \to \varphi \left( {\vec x,y + 1} \right)} \right)} \right) \to \forall z\varphi \left( {\vec x,z} \right)} \right), \) where a formula is called bounded if it contains only quantifiers of the type \( \forall x < y{\text{ }}or{\text{ }}\exists x < y. \)

Keywords

Natural Number Boolean Formula Proper Solution Isomorphism Type Peano Arithmetic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [Ajl]
    M. Ajtai, Firstorder definability on finite structures, Annals of Pure and Applied Logic 45 (1989) 211–225.CrossRefGoogle Scholar
  2. [Aj2]
    M. Ajtai, The complexity of the Pigeonhole Principle 29-th, Annual Symposium on Foundations of Computer Science, 1988, 346–358.Google Scholar
  3. [CR]
    S. Cook and R. Rechkhow, The relative efficiency of propositional proof systems, Journal of Symbolic Logic 44 (1977) 36–50.Google Scholar
  4. [PW]
    J.B. Paris and A.J. Wilkie, Counting Problems in Bounded Arithmetic, in Methods in Mathematical Logic, Proc. Caracas 1983, Springer- Verlag Lecture Notes in Mathematics no. 1130. Ed: A. Dold and B. Eckrnan, Springer-Verlag, 1985, pp. 317–340.CrossRefGoogle Scholar
  5. [PWW]
    J.B. Paris and A.J. Wilkie, Provability of the pigeonhole principle and the existence of infinitely many primes. Journal of Symbolic Logic 53 (1988) 1235–1244.CrossRefGoogle Scholar
  6. [Wi]
    A.J. Wilkie, talk presented at the ASL Summer meeting in Manchester, England,. 1984.Google Scholar
  7. [Wo]
    A.R. Woods, Some problems in logic and number theory and their connections, Ph.D. dissertation, Department of Mathematics, Manchester University, 1981.Google Scholar

Copyright information

© Birkhäuser Boston 1990

Authors and Affiliations

  • M. Ajtai
    • 1
  1. 1.IBM Almaden Research CenterSan JoseUSA

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