Herbrand Consistency of Some Arithmetical Theories

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, $\mathit{\text{Fundamenta}}\mathit{\text{Mathematicae}}$ 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 $\mathrm{I}{\mathrm{\Delta }}_0+{\mathrm{\Omega }}_{\mathrm{m}}$ 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 $T\supseteq \mathrm{I}{\mathrm{\Delta }}_0+{\mathrm{\Omega }}_2$ 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 $\mathrm{I}{\mathrm{\Delta }}_0$ 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 $\mathrm{I}{\mathrm{\Delta }}_0+{\mathrm{\Omega }}_1$ and ${\rm I\Delta_0}$.