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Herbrand Consistency of Some Finite Fragments of Bounded Arithmetical Theories (1110.1848v2)
Published 9 Oct 2011 in math.LO and cs.LO
Abstract: We formalize the notion of Herbrand Consistency in an appropriate way for bounded arithmetics, and show the existence of a finite fragment of ${\rm I\Delta_0}$ whose Herbrand Consistency is not provable in the thoery ${\rm I\Delta_0}$. We also show the existence of an ${\rm I\Delta_0}-$derivable $\Pi_1-$sentence such that ${\rm I\Delta_0}$ cannot prove its Herbrand Consistency.