Logic Colloquium 2008

Roman, KossakAutomorphisms of models of PA: the neglected casesThe Graduate Center, City University of New York, 365 Fifth Avenue, 10016 New York, USA rkossak@gc.cuny.edu

Automorphism groups of countable recursively saturated models of PA were studied intensively in the 1990's. In a series of papers (joint and separate) by Lascar, Kaye, Kotlarski, Schmerl and the author, the important notion of arithmetic saturation was isolated and a series of results characterizing arithmetic saturation in group theoretic terms were proved. The notions involved include: generic automorphisms, maximal automorphisms, the small index property, maximal open subgroups, and the cofinality of the group. Most of this material is presented in [1]. Much less has been done concerning models which are recursively saturated and not arithmetically saturated (we do not even have a good name for this class of models), and even less concerning short recursively saturated models. I will give a survey of this entire area of research and I will include some more recent results concerning the neglected cases, in particular the problem of extendabability of automorphisms to cofinal extensions. This last topic is joint work with Henryk Kotlarski.


  • 1]
    Roman Kossak, James Schmerl, The Structure of Models of Peano Arithmetic, Oxford Logic Guides 50, Oxford University Press, 2006.