On two problems concerning end extensions (Q937209)

The paper presents a comprehensive survey and some new results concerning two questions on end extensions of models of fragments of PA. The first, due to Paris, asks whether every countable model of \(\text{B}\Sigma_1\) has a proper end extension to a model of \(\text{I}\Delta_0\); the second, due to Clote, asks whether every countable model of \(\text{B}\Sigma_n\), \(n\geq 2\), has a proper \(\Sigma_n\)-elementary end extension to a model of \(\text{B}\Sigma_{n-1}\). Concerning the first question, \textit{A. Wilkie} and \textit{J. Paris} proved in [``On the existence of end extensions of models of bounded induction'', in: J. E. Fenstad et al. (eds.), Logic, methodology and philosophy of science VIII, Proc. 8th Int. Congr., Moscow/USSR 1987, Stud. Logic Found. Math. 126, 143--161 (1989; Zbl 0695.03019)] that the answer is affirmative if the model is \(\Delta_0\)-full. Cornaros and Dimitracopoulos analyze this notion and relate it to other notions of weak saturation. They introduce a notion of \(\Gamma\)-1-fullness, where \(\Gamma\) is a class of formulas, and they prove that if \(\Gamma\supseteq \text{I}\Delta_0\) is a recursive set of sentences and \(M\) is a countable model of \(\text{B}\Sigma_1+\Pi_1(\Gamma)\) and there is a sentence \(\lambda\) such that \(M\models\lnot\lambda\) and \(M\) is \((\Gamma+\lambda)\)-1-full, then there exists \(K\models\Gamma\) which is a proper \(\Sigma_1\)-elementary end extension of \(M\). They also prove that if there is a \(K\) as above and \(\Gamma\supseteq \text{I}\Sigma_1\), then \(M\) is \(\Gamma\)-1-full. Concerning Clote's question, Cornaros and Dimitracopoulos prove that for \(n\geq 2\) every countable model of \(\text{B}\Sigma_n\) has a proper \(\Sigma_n\)-elementary end extension to a model of parameter-free collection scheme for \(\Sigma_{n-1}\) formulas. They also modify a proof from the above mentioned paper of Wilkie and Paris to show that for every \(n\geq 2\) there exists a countable \(M\models \text{B}\Sigma_n\) for which there is no proper \(\Sigma_{n-1}\)-elementary end extension to a model of \(\text{I}\Sigma_{n-1}\).