Herbrand consistency of some arithmetical theories

Abstract

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+Ω_m 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.