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Abstract

We prove that the Gödel incompleteness theorem holds for a weak arithmetic $T_m=I \Delta _0+ \Omega _m$, for $m\ge 2$, in the form $T_m\not \vdash {\rm HCons}(T_m)$, where ${\rm HCons}(T_m)$ is an arithmetic formula expressing the consistency of $T_m$ with respect to the Herbrand notion of provability. Moreover, we prove $T_m\not \vdash {\rm HCons}^{I_m}(T_m)$, where ${\rm HCons}^{I_m}$ is ${\rm HCons}$ relativised to the definable cut $I_m$ of $(m-2)$-times iterated logarithms. The proof is model-theoretic. We also prove a certain non-conservation result for $T_m$.