Categoricity, amalgamation, and tameness
The categoricity problem in \(L_{\omega_1, \omega}\) is considered. For every \(k\) with \(2 \leq k < \omega\), an \(L_{\omega_1, \omega}\)-sentence \(\varphi_k\) is built satisfying the following conditions: (1) \(\varphi_k\) is categorical in \(\mu\) when \(\mu \leq \aleph_{k-2}\) but not when \(\mu > \aleph_{k-2}\); (2) \(\varphi_k\) is not Galois stable in \(\aleph_{k-2}\); (3) \(\varphi_k\) has the disjoint amalgamation property. Moreover, for \(k > 2\), (4) \(\varphi_k\) is \((\aleph_0, \, \aleph_{k-3})\)-tame, and indeed first-order types determine Galois types over models of cardinality \(\leq \aleph_{k-3}\); (5) \(\varphi_k\) is Galois-stable in \(\aleph_m\) for \(m \leq k-3\); (6) \(\varphi_k\) is not \((\aleph_{k-3}, \, \aleph_{k-2})\)-tame. Also, it is shown that models of \(\varphi_k\) are a model-complete abstract elementary class AEC. The construction refers to earlier work of Hart and Shelah, but adds new significant contributions with respect to amalgamation, tameness and stability. In particular it provides a new example of non-tame AEC (simpler than that previously given by the first author and Shelah).
scientific article; zbMATH DE number 5601722
scientific article