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

Separating Bounded Arithmetics by Herbrand Consistency