Herbrand Consistency of Some Arithmetical Theories

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 171 (2002) 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 I∆0 + 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 ⊇ I∆0 + Ω2 in T itself. In this paper, the above results are generalized for I∆0 + Ω1. Also after tailoring the definition of Herbrand consistency for I∆0 we prove the corresponding theorems for I∆0. Thus the Herbrand version of Gödel’s second incompleteness theorem follows for the theories I∆0 + Ω1 and I∆0. vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv 2010 Mathematics Subject Classification: Primary 03F40, 03F30; Secondary 03F05, 03H15.