Herbrand consistency and bounded arithmetic (Q2773372)

It is proved that Gödel's incompleteness theorem holds for a weak arithmetic \(T_{m} = I \Delta_{0} + \Omega_{m}\) for \(m \geq 2\) in the form \(T_{m} \nvdash \text{HCons}(T_{m})\) where \(\text{HCons}(T_{m})\) is an arithmetical formula expressing the consistency of \(T_{m}\) with respect to the Herbrand notion of provability. It is also proved that \(T_{m} \nvdash \text{HCons}^{I_{m}}(T_{m})\) where \(\text{HCons}^{I_{m}}\) is HCons relativised to the definable cut \(I_{m}\) of \((m-2)\)-times iterated logarithms. The theorems are proved by model-theoretic methods. In the paper a certain non-conservation result for \(T_{m}\) is also proved.