Separating Bounded Arithmetics by Herbrand Consistency

TL;DR

This paper investigates the limitations of Herbrand Consistency in separating bounded arithmetic theories, showing it cannot distinguish certain hierarchies but can separate others, extending previous results on unprovability.

Contribution

It proves that Herbrand Consistency cannot $ ext{Pi}_1$-separate specific bounded arithmetic theories, extending earlier work on unprovability within these theories.

Findings

Herbrand Consistency cannot $ ext{Pi}_1$-separate ${ m I riangle_0+igwedge_jigcirc_j}$ from ${ m I riangle_0}$

Herbrand Consistency can $ ext{Pi}_1$-separate ${ m I riangle_0+Exp}$ from ${ m I riangle_0}$

Unprovability of Herbrand Consistency of ${ m I riangle_0}$ in ${ m I riangle_0+igwedge_jigcirc_j}$

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