Existentially closed structures and Gödel's second incompleteness theorem (Q2732285)

A model \(M\) of a theory \(T\) is 1-closed if for every \(M'\models T\) if \(M\prec_{\Sigma_0}M'\) then \(M\prec_{\Sigma_1}M'\). Since Matiyasevich's theorem is provable in \(I\Delta_0+\text{exp}\), every existentially closed model of a theory \(T\supseteq I\Delta_0+\text{exp}\) is 1-closed. In the first part of the paper is it shown that if \(T\supseteq I\Delta_0+\text{exp}\), \(T\subseteq T'\), and \(M\) is a 1-closed model of \(T\), then \(M\models\lnot \text{Cons}(T')\). The proof is model-theoretic and makes no \text{exp}licit (nor implicit) use of the diagonal lemma. In the second part similar ideas and a version of the arithmetized completeness theorem are used to prove the formalized version of the second incompleteness theorem: \(\text{PA}\vdash \text{Cons}(\text{PA})\rightarrow \text{Cons}(\text{PA}+\lnot \text{Cons}(\text{PA}))\).