On Maximal Theories

Let S be a recursive theory. Let a theory T * consisting of Σ 1 sentences be called maximal (with respect to S ) if T * is maximal consistent with S , i.e. there is no Σ 1 sentence consistent with T * + S which is not in T *. A maximal theory with respect to I Δ 0 was considered by Wilkie and Paris in [WP] in connection with the end-extension problem. Let us recall that I Δ 0 is the fragment of Peano arithmetic consisting of the finite collection of algebraic axioms PA − together with the induction scheme restricted to bounded formulas. The main open problem concerning the end-extendability of models of I Δ 0 is the following: (*) Does every model of I Δ 0 + B Σ 1 have a proper end-extension to a model of I Δ 0 ? Here B Σ 1 is the following collection scheme: where φ runs over bounded formulas and may contain parameters. It is well known(see [KP]) that if I is a proper initial segment of a model M of I Δ 0 , then I satisfies I Δ 0 + B Σ 1 . For a wide discussion of the problem (*) see [WP]. Wilkie and Paris construct in [WP] a model M of I Δ 0 + B Σ 1 which has no proper end-extension to a model of I Δ 0 under the assumption I Δ 0 ⊢¬Δ 0 H (see [WP] for an explanation of this assumption). Their model M is a model of a maximal theory T * where S = I Δ 0 . Moreover, T *, which is the set Σ 1 ( M ) of all Σ 1 sentences true in M , is not codable in M .