Lower and upper bounds for the provability of Herbrand consistency in weak arithmetics
Volume 212 / 2011
Fundamenta Mathematicae 212 (2011), 191-216
MSC: Primary 03F30; Secondary 03F40.
DOI: 10.4064/fm212-3-1
Abstract
We prove that for $i\geq 1$, the arithmetic ${\rm I}\Delta_0 + \Omega_i$ does not prove a variant of its own Herbrand consistency restricted to the terms of depth in $(1+\varepsilon)\log^{i+2}$, where $\varepsilon$ is an arbitrarily small constant greater than zero. On the other hand, the provability holds for the set of terms of depths in $\log^{i+3}$.
