Herbrand Consistency of Some Finite Fragments of Bounded Arithmetical Theories

We formalize the notion of Herbrand Consistency in an appropriate way for bounded arithmetics, and show the existence of a finite fragment of ${\rm I\Delta_0}$whoseHerbrandConsistencyisnotprovableinthethoery${\rm I\Delta_0}$\mathrm{.}Wealsoshowtheexistenceofan${\rm I\Delta_0}-$derivable$\Pi_1-$sentencesuchthat${\rm I\Delta_0}$ cannot prove its Herbrand Consistency.