Automorphisms of models of true arithmetic: Recognizing some basic open subgroups. (English) Zbl 0824.03016

The study of the automorphism group of a structure \(M\) (whether viewed as an abstract group, a permutation group acting on \(M\), or even a topological group with sub-basic open sets of the form \(U^ b_ a= \{g\in \operatorname{Aut}(M): g(a)= b\})\) and the amount of information it conveys concerning \(M\) itself has been of considerable recent interest [see, e.g., Automorphisms of first-order structures, edited by D. Macpherson and the second author (1994; Zbl 0797.00010)]. This study has been proved particularly fruitful when \(M\) is a countable recursively saturated model of PA. The paper under review focuses on models of the most distinguished completion of PA, namely true arithmetic (the truth- set of the standard model of PA). The main problem in this area is to distinguish the basic subgroups of \(\operatorname{Aut}(M)\) (those of the form \(U^ a_ a\)), viewed as a topological group, from the open ones by a purely topological-group-theoretic property.
The authors give properties of this sort, which are satisfied by subclasses of the basic subgroups, namely strongly maximal and maximal ones. The problem of finding a property describing precisely the basic subgroups remains unsolved, but Kossak and Schmerl have recently solved this problem for the special case of arithmetically saturated models (models in which the types which are finitely satisfiable and arithmetical in some set in the standard system of the model are realized, or, equivalently, recursively saturated models in which the standard cut \(\mathbb{N}\) is strong).


03C62 Models of arithmetic and set theory
20B27 Infinite automorphism groups


Zbl 0797.00010
Full Text: DOI


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