Herbrand Consistency of Some Finite Fragments of Bounded Arithmetical Theories

TL;DR

This paper investigates the provability of Herbrand Consistency within bounded arithmetical theories, demonstrating limitations in ${ m I riangle_0}$ by constructing specific unprovable instances.

Contribution

It formalizes Herbrand Consistency for bounded arithmetics and proves the existence of finite fragments where Herbrand Consistency is unprovable in ${ m I riangle_0}$.

Findings

Existence of a finite ${ m I riangle_0}$ fragment with unprovable Herbrand Consistency

Existence of a ${ m I riangle_0}$-derivable $orall ext{Pi}_1$ sentence with unprovable Herbrand Consistency in ${ m I riangle_0}$

Demonstrates limitations of ${ m I riangle_0}$ in proving certain consistency statements.

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.