Separating Bounded Arithmetics by Herbrand Consistency

Abstract: The problem of $\Pi_1-$separating the hierarchy of bounded arithmetic has been studied in the paper. It is shown that the notion of Herbrand Consistency, in its full generality, cannot $\Pi_1-$separate the theory ${\rm I\Delta_0+\bigwedge_j\Omega_j}$ from ${\rm I\Delta_0}$; though it can $\Pi_1-$separate ${\rm I\Delta_0+Exp}$ from ${\rm I\Delta_0}$. This extends a result of L. A. Ko{\l}odziejczyk (2006), by showing the unprovability of the Herbrand Consistency of ${\rm I\Delta_0}$ in the theory ${\rm I\Delta_0+\bigwedge_j\Omega_j}$.