Automorphisms of saturated and boundedly saturated models of arithmetic.

*(English)*Zbl 1241.03046The paper is devoted to the study of automorphism groups of saturated and boundedly saturated models of Peano arithmetic. A model is boundedly saturated if it is saturated with respect to bounded types. A type over a model \(M\) is bounded if it contains the formula \(x<a\) for some \(a\in M\). The authors prove several results analogous to previously known theorems concerning automorphisms and automorphism groups of countable recursively saturated models of PA. Some results are straightforward corollaries of corresponding statements about countable recursively saturated models, some require new alternative approaches. The authors prove an analog of a theorem of C. Smoryński [“Back-and-forth inside a recursively saturated model of arithmetic”, Stud. Logic Found. Math. 108, 273–278 (1982; Zbl 0503.03033)] and a theorem of R. Kaye [“A Galois correspondence for countable recursively saturated models of Peano arithmetic”, in: Automorphisms of first-order structures. Oxford: Clarendon Press. 293–312 (1994; Zbl 0824.03015)].

Let \(M\) be a boundedly saturated model of PA of cardinality \(\lambda\), and let \(G\) be its automorphism group. The first theorem states that a cut \(I\) of \(M\) is the largest cut fixed pointwise by some \(f\) in \(G\) iff \(I\) is closed under exponentiation and every downward cofinal subset of \(M\setminus I\) has cardinality smaller than \(\lambda\). The second theorem says that a closed subgroup of \(G\) is normal iff it is the pointwise stabilizer of an invariant cut of \(M\). The basis of the topology on \(G\) consists of all pointwise stabilizers of finite sets and their cosets. Among other results, it is shown that if \(M_1\) and \(M_2\) are saturated, \(M_1\) is an elementary extension of the standard model and \(M_2\) is not, then \(\text{Aut}(M_1)\not\cong\text{Aut}(M_2)\), and that every saturated model has short elementary cuts whose automorphism groups are nonisomorphic.

Let \(M\) be a boundedly saturated model of PA of cardinality \(\lambda\), and let \(G\) be its automorphism group. The first theorem states that a cut \(I\) of \(M\) is the largest cut fixed pointwise by some \(f\) in \(G\) iff \(I\) is closed under exponentiation and every downward cofinal subset of \(M\setminus I\) has cardinality smaller than \(\lambda\). The second theorem says that a closed subgroup of \(G\) is normal iff it is the pointwise stabilizer of an invariant cut of \(M\). The basis of the topology on \(G\) consists of all pointwise stabilizers of finite sets and their cosets. Among other results, it is shown that if \(M_1\) and \(M_2\) are saturated, \(M_1\) is an elementary extension of the standard model and \(M_2\) is not, then \(\text{Aut}(M_1)\not\cong\text{Aut}(M_2)\), and that every saturated model has short elementary cuts whose automorphism groups are nonisomorphic.

Reviewer: Roman Kossak (New York)