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# Parity and the Pigeonhole Principle

## 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* Δ_{0} 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## Preview

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## References

- [Ajl]M. Ajtai, Firstorder definability on finite structures, Annals of Pure and Applied Logic 45 (1989) 211–225.CrossRefGoogle Scholar
- [Aj2]M. Ajtai, The complexity of the Pigeonhole Principle 29-th, Annual Symposium on Foundations of Computer Science, 1988, 346–358.Google Scholar
- [CR]S. Cook and R. Rechkhow, The relative efficiency of propositional proof systems, Journal of Symbolic Logic 44 (1977) 36–50.Google Scholar
- [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
- [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
- [Wi]A.J. Wilkie, talk presented at the ASL Summer meeting in Manchester, England,. 1984.Google Scholar
- [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