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.