Automorphism groups of countable arithmetically saturated models of Peano arithmetic (Q2795926)

The paper is devoted to the study of automorphism groups of models of arithmetic. The main results of it are connected with the following question: to what extent (the isomorphism type of) the group \(\mathrm{Aut}(\mathcal M)\) of all automorphisms of a countable recursively saturated model \(\mathcal M\) of Peano arithmetic PA determines (the isomorphism type of) \(\mathcal M\)? The principal result states that if \(\mathcal M\), \(\mathcal N\) are countable, arithmetically saturated models of PA and \(\mathrm{Aut}(\mathcal M) \simeq \mathrm{Aut} (\mathcal N)\) then the Turing-jumps of \(\mathrm{Th}(\mathcal M)\) and \(\mathrm{Th}(\mathcal N)\) are recursively equivalent. It is also proved that for each \(n < \omega\) there are recursively equivalent completions \(T_{0}, T_{1}, \ldots, T_{n}\) of PA such that whenever \(i < j \leq n\) and \(\mathcal M_i\), \(\mathcal M_j\) are countable arithmetically saturated models of \(T_{i}\) and \(T_{j}\), respectively, then \(\mathrm{Aut}(\mathcal M_i) \not\simeq \mathrm{Aut}(\mathcal M_j)\).