Gödel's second incompleteness theorem is proved for Herbrand consistency of some arithmetical theories with bounded induction, by using a technique of logarithmic shrinking the witnesses of bounded formulas, due to Z. Adamowicz [Herbrand consistency and bounded arithmetic, Fundamenta Mathematicae, vol. 171 (2002), pp. 279—292]. In that paper, it was shown that one cannot always shrink the witness of a bounded formula logarithmically, but in the presence of Herbrand consistency, for theories IΔ0+Ωm with m ≥ 2, any witness for any bounded formula can be shortened logarithmically. This immediately implies the unprovability of Herbrand consistency of a theory T⊇ IΔ0+Ω2 in T itself.
In this paper, the above results are generalized for IΔ0+Ω1. Also after tailoring the definition of Herbrand consistency for IΔ0 we prove the corresponding theorems for IΔ0. Thus the Herbrand version of Gödel's second incompleteness theorem follows for the theories IΔ0+Ω1 and IΔ0.
"Herbrand consistency of some arithmetical theories." J. Symbolic Logic 77 (3) 807 - 827, September 2012. https://doi.org/10.2178/jsl/1344862163