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An arithmetic bounded formula uses only quantifiers bounded by arithmetic terms, such as $\exists u<t$ and $\forall u<t$, with the bound not containing the quantified variable. Existential bounded representation of a primitive recursive function places all unrestricted existential variables in a single outer block.

A bounded arithmetic sentence is an arithmetic bounded formula with no free variables. Every quantifier is restricted by an arithmetic first-order term. In the standard natural numbers such a first-order sentence has a definite truth value determined by finite bounded searches.