Amalgamation, Absoluteness, and Categoricity

We describe the major result on categoricity in Lω1,ω , placing it in the context of more general work in abstract elementary classes. In particular, we illustrate the role of higher dimensional amalgamations and sketch the role of a weak extension of ZFC in the proof. We expound the translation of the problem to studying atomic models of first order theories. We provide a simple example of the failure of amalgamation for a complete sentence of Lω1,ω . We prove some basic results on the absoluteness of various concepts in the model theory of Lω1,ω and publicize the problem of absoluteness of א1-categoricity in this context. Stemming from this analysis, we prove Theorem: The class of countable models whose automorphism groups admit a complete left invariant metric is Π1 but not Σ1. The study of infinitary logic dates from the 1920’s. Our focus here is primarily on the work of Shelah using stability theoretic methods in the field (beginning with [She75]). In the first four sections we place this work in the much broader context of abstract elementary classes (aec), but do not develop that subject here. The main result discussed, Shelah’s categoricity transfer theorem for Lω1,ω explicitly uses a weak form of the GCH. This raises questions about the absoluteness of fundamental notions in infinitary model theory. Sections 5-7 and the appendix due to David Marker describe the complexity and thus the absoluteness of such basic notions as satisfiability, completeness, ω-stability, and excellence.1 We state the question, framed in this incisive way by Laskowski, of the absoluteness of א1-categoricity. And from the model theoretic characterization of non-extendible models we derive the theorem stated in the abstract on the complexity of automorphism groups. Most of the results reported here in Sections 1-4 are due to Shelah; the many references to [Bal09] are to provide access to a unified exposition. I don’t know anywhere that the results in Section 5 have been published; but the techniques are standard and our goal is just to provide a reference. The result in Section 6 is new but easy. ∗This article is a synthesis of the paper given in Singapore with later talks, including the Mittag-Leffler Institute in 2009 and CRM Barcelona in 2010. It reflects discussions with set theorists during my stay at Mittag-Leffler and discussion with the Infinity project members in Barcelona. The author wishes to thank the John Templeton Foundation for its support through Project #13152, Myriad Aspects of Infinity, hosted during 2009-2011 at the Centre de Recerca Matematica, Bellaterra, Spain. Baldwin was partially supported by NSF-0500841. 1David Marker is partially supported by National Science Foundation grant DMS-0653484.