# 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 $$\forall x < y{\text{ }}or{\text{ }}\exists x < y.$$

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