We prove that the Gödel incompleteness theorem holds for a weak arithmetic Tₘ = IΔ₀ + Ωₘ, for m ≥ 2, in the form Tₘ ⊬ HCons(Tₘ), where HCons(Tₘ) is an arithmetic formula expressing the consistency of Tₘ with respect to the Herbrand notion of provability. Moreover, we prove $Tₘ⊬HCon{s}^{Iₘ}\left(Tₘ\right)$, where $HCon{s}^{Iₘ}$ is HCons relativised to the definable cut Iₘ of (m-2)-times iterated logarithms. The proof is model-theoretic. We also prove a certain non-conservation result for Tₘ.