A sharp version of the bounded Matijasevich conjecture and the end-extension problem

Abstract

Under a sharp version of the assumption I∆0 ⊢ ¬ ∆0H we characterize models of I∆0 + BΣ1 having a proper end extension to a model of I∆0.In [WP] Wilkie and Paris study the relationship between the existence of a proper end extension of a model and its “fullness”, which is related to a certain weak overspill principle (we recall the definition of fullness in §3). Let M be a countable nonstandard model of I∆0 + BΣ1. Under the hypothesis I∆0 ⊢ ¬ ∆0H they prove the following (Corollaries 7 and 8):The following are equivalent:1) M has a proper end extension to a model of I∆0 + BΣ1.2) M is (I∆0 + BΣ1)-full.Moreover, assuming that there is no t ∈ M such that for v ∈ M, 2[t/v]exists if and only if v < N, the following are equivalent:1) M has a proper end extension to a model of I∆0.2) M is I∆0-full.Wilkie and Paris ask whether the assumption on the structure of the model can be eliminated from the second equivalence.We eliminate it, but we sharpen the assumption I∆0 ⊢ ¬ ∆0H. So we partially answer their question.