Herbrand Consistency of Some Finite Fragments of Bounded Arithmetical Theories

Abstract

We formalize the notion of Herbrand Consistency in an appropriate way for bounded arithmetics, and show the existence of a finite fragment of \({\rm I\Delta_0}\) whose Herbrand Consistency is not provable in the thoery \({\rm I\Delta_0}\). We also show the existence of an \({\rm I\Delta_0}-\)derivable \(\Pi_1-\)sentence such that \({\rm I\Delta_0}\) cannot prove its Herbrand Consistency.