A note on Σ1-maximal models

Abstract

Let T be a recursive theory in the language of first order Arithmetic. We prove that if T extends: (a) the scheme of parameter free Δ₁-minimization (plus exp), or (b) the scheme of parameter free Π₁-induction, then there are no Σ₁-maximal models with respect to T. As a consequence, we obtain a new proof of an unpublished theorem of Jeff Paris stating that Σ₁-maximal models with respect to IΔ₀ + exp do not satisfy the scheme of Σ₁-collection BΣ₁.