Minimal satisfaction classes with an application to rigid models of Peano arithmetic (Q1182707)

In this paper \(\omega_ 1\)-like recursively saturated models of Peano Arithmetic without nontrivial automorphisms and moreover without nontrivial elementary embeddings into itself are considered. Models which have no nontrivial automorphisms are called rigid. A satisfaction class \(S\) for a model \(M\) is called minimal if \((M,S)\) has no proper elementary substructures. Let \(Q_ n\) denote the closure of a certain set of formulas in any language \(L\) of Peano Arithmetic under negation, conjunction and bounded quantification. \(Q_ \infty\) is the set of all formulas of \(L\). Then one obtains: Let \(M\) be a nonstandard countable model of Peano Arithmetic and let \(S\) be a \(Q_ e\)-class for \(M\), where either \(e\in M\) or \(e=\infty\). Then \(M\) has continuum many pairwise inequivalent \(Q_ e\)-classes \(D\) such that \(D|_ d = S|_ d\) and \((M,D,d)\) is minimal for any \(d\) with \(d+\omega<e\). Every countable, recursively saturated model of Peano Arithmetic has an \(\omega_ 1\)-like, recursively saturated, elementary end extension without nontrivial elementary embeddings into itself. This assertion is true for all uncountable regular cardinals \(\kappa >| M|\), too, if \(M\) has countable cofinality.