On the Herbrand notion of consistency for finitely axiomatizable fragments of bounded arithmetic theories

Modifying the methods of Z. Adamowicz’s paper Herbrand consistency and bounded arithmetic (Fund. Math. 171 (2002)), we show that there exists a number n such that ⋃ m Sm (the union of the bounded arithmetic theories Sm) does not prove the Herbrand consistency of the finitely axiomatizable theory Sn 3 . From the point of view of bounded arithmetic, the concept of consistency based on Herbrand’s theorem has at least two interesting features. Firstly, it has a more combinatorial flavour than standard consistency notions, and thus lends itself more naturally to combinatorial interpretations (cf. [Pud]). Secondly, it seems reasonably weak. It is well-known that for stronger concepts of consistency, such as ordinary Hilbertor Gentzen-style consistency (Cons), or even bounded consistency (consistency with respect to proofs containing only bounded formulae, BdCons), there is typically a very large gap between a given theory and the theories whose consistency it can prove. Wilkie and Paris ([WP87]) ∗Part of this work was carried out while the author was a Foundation for Polish Science (Fundacja na rzecz Nauki Polskiej) scholar.