On closed elementary cuts in recursively saturated models of Peano arithmetic (Q1309332)

If \(M\) is a model of PA and \(a\in M\), then \(M[a]\) is the largest elementary submodel of \(M\) that does not contain \(a\). A subset \(X\) of \(M\) is closed if for every \(b\not\in M\) there is an automorphism of \(M\) that fixes \(X\) pointwise and moves \(b\). \textit{H. Kotlarski} has proved in ``On elementary cuts in recursively saturated models of Peano Arithmetic'' [Fundam. Math. 120, 205-222 (1984; Zbl 0572.03016)] that every countable recursively saturated model of PA has closed elementary cuts, and that all such cuts have to be of the form \(M[b]\). In the same paper Kotlarski proved that for every countable recursively saturated model \(M\) there are countably many nonisomorphic structures of the form \((M,M[b])\). The cuts \(M[b]\) that Kotlarski used in his proof were not closed. In the paper under review a refinement of Kotlarski's technique is applied to prove that for every \(M\) as above there are countably many nonisomorphic structures of the form \((M,M[b])\), where \(M[b]\) is closed in \(M\).