  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 ${\rm
I\Delta_0+\bigwedge_j\Omega_j}$.
