  G\"odel's second incompleteness theorem is proved for Herbrand consistency of
some arithmetical theories with bounded induction, by using a technique of
logarithmic shrinking the witnesses of bounded formulas, due to Z. Adamowicz
[Herbrand consistency and bounded arithmetic, \textit{Fundamenta Mathematicae}
171 (2002) 279--292]. In that paper, it was shown that one cannot always shrink
the witness of a bounded formula logarithmically, but in the presence of
Herbrand consistency, for theories ${\rm I\Delta_0+\Omega_m}$ with $m\geqslant
2$, any witness for any bounded formula can be shortened logarithmically. This
immediately implies the unprovability of Herbrand consistency of a theory
$T\supseteq {\rm I\Delta_0+\Omega_2}$ in $T$ itself.
  In this paper, the above results are generalized for ${\rm
I\Delta_0+\Omega_1}$. Also after tailoring the definition of Herbrand
consistency for ${\rm I\Delta_0}$ we prove the corresponding theorems for ${\rm
I\Delta_0}$. Thus the Herbrand version of G\"odel's second incompleteness
theorem follows for the theories ${\rm I\Delta_0+\Omega_1}$ and ${\rm
I\Delta_0}$.
