Automorphisms of countable short recursively saturated models of PA (Q1049737)

Let \(M\) be a countable recursively saturated model of PA. For each \(a\in M\), let \(M(a)\) be the smallest elementary initial segment of \(M\) which contains \(a\). \(M(a)\) is not recursively saturated, but it realizes all bounded consistent recursive types with parameters from \(M\); hence \(M(a)\) is called \textit{short recursively saturated}. Much is known about automorphisms and automorphism groups of recursively saturated models of PA. Shochat investigates how much of it can be reproved in the short saturated case. Some results translate directly, some require new proofs, and there is a list of interesting open problems. The main part of the paper is devoted to the two groups: \(G(a)=\text{Aut}(M(a))\) and its subgroup \(G|_{M(a)}\) whose elements are those \(g\in G(a)\) which extend to an automrphism of \(M\). The main results are that the index \(G|_{M(a)}\) in \(G(a)\) is \(2^{\aleph_0}\), and that \(G|_{M(a)}\) is not a normal subgroup of \(G(a)\).