Separating bounded arithmetical theories by Herbrand consistency

The problem of Π1–separating the hierarchy of bounded arithmetic has been studied in the article. It is shown that the notion of Herbrand consistency, in its full generality, cannot Π1–separate the theory IΔ0+⋀jΩj from IΔ0; though it can Π1–separate IΔ0+Exp from IΔ0. Namely, we show the unprovability of the Herbrand consistency of IΔ0 in the theory IΔ0+⋀jΩj. This partially extends a result of L. A. Kołodziejczyk who showed that for a finite fragment S⊆IΔ0, the Herbrand consistency of S+Ω1 is not provable in IΔ0+⋀jΩj.