Separating Bounded Arithmetics by Herbrand Consistency

Abstract

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 $\mathrm{I}{\mathrm{\Delta }}_0+{\bigwedge }_{\mathrm{j}}{\mathrm{\Omega }}_{\mathrm{j}}$ from $\mathrm{I}{\mathrm{\Delta }}_0$; though it can ${\Pi }_1-$separate $\mathrm{I}{\mathrm{\Delta }}_0+\mathrm{Exp}$ from $\mathrm{I}{\mathrm{\Delta }}_0$. This extends a result of L. A. Kołodziejczyk (2006), by showing the unprovability of the Herbrand Consistency of $\mathrm{I}{\mathrm{\Delta }}_0$ in the theory $\mathrm{I}{\mathrm{\Delta }}_0+{\bigwedge }_{\mathrm{j}}{\mathrm{\Omega }}_{\mathrm{j}}$.

Published by Oxford University Press. arXiv admin note: text overlap with arXiv:1005.2654

References in corpus (1)

• Herbrand Consistency of Some Arithmetical Theories

Cited by in corpus (2)

• Herbrand Consistency of Some Arithmetical Theories
• Herbrand Consistency of Some Finite Fragments of Bounded Arithmetical Theories