Separating bounded arithmetical theories by Herbrand consistency (Q2893322)

The paper is devoted to the problem of \(\Pi_{1}\)-separating the hierarchy of bounded arithmetic. It is shown that the notion of Herbrand consistency in its full generality cannot \(\Pi_{1}\)-separate the theory \(\mathrm{I}\Delta_{0} + \bigwedge_{j} \Omega_{j}\) from \(\mathrm{I}\Delta_{0}\), though it can \(\Pi_{1}\)-separate \(\mathrm{I}\Delta_{0} + \mathrm{Exp}\) from \(\mathrm{I}\Delta_{0}\). In fact it is proved that Herbrand consistency of \(\mathrm{I}\Delta_{0}\) cannot be proved in the theory \(\mathrm{I}\Delta_{0} + \bigwedge_{j} \Omega_{j}\). The result partially extends a result of L. A. Kołodziejczyk, who showed that for a finite fragment \(S \subseteq \mathrm{I}\Delta_{0}\) the Herbrand consistency of \(S + \Omega_{1}\) is not provable in \(\mathrm{I}\Delta_{0} + \bigwedge_{j} \Omega_{j}\).