Lower and upper bounds for the provability of Herbrand consistency in weak arithmetics

摘要

We prove that for i ≥ 1, the arithmetic IΔ_0 + Ω_i does not prove a variant of its own Herbrand consistency restricted to the terms of depth in (1 + ?)log~(i+2), where e 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).