# Parity and the Pigeonhole Principle

• M. Ajtai
Chapter
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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