Categoricity of theories in \(L_{\kappa \omega}\), with \(\kappa\) a compact cardinal
In this paper the authors generalize the classical Morley categoricity theorem to a particular kind of infinitary languages, \(L_{\kappa \omega}\) with \(\kappa\) a compact cardinal. One of the main results of the paper is the following Theorem: Let \(\kappa\) be an uncountable strongly compact cardinal, \(T\) a theory in a fragment \(F\) of \(L_{\kappa \omega}\) and \(\kappa '=\max (\kappa,| F|)\). Let \(\lambda\) be a successor cardinal with \(\lambda >((\kappa ')^{<\kappa})^+\). If \(T\) is categorical in \(\lambda\) then \(T\) is categorical in every cardinal greater or equal to \(\min (\lambda,\beth_{(2^{\kappa '})^+})\). This paper should be seen as part of the program of classification theory. In Section 1 the necessary preliminaries are given, amalgamation and joint embedding property are investigated. In Section 2 types are studied under the assumption of amalgamation and joint embedding property. In Section 3 the authors deal with order indiscernibles and introduce the Skolem hull. In Section 4 an extension of an elementary part of stability is given. A suitable notion of non-forking over models is developed and used to obtain the categoricity result. Section 5 gives a summary. An appendix contains set-theoretic material used in Section 1. It gives the necessary background concerning modified square-systems.
scientific article; zbMATH DE number 4154417
scientific article