Categoricity, Amalgamation, and Tameness

Theorem: For each 2 ≤ k < ω there is an Lω1,ω -sentence ϕk such that (1) ϕk is categorical in μ if μ≤ℵk−2; (2) ϕk is not ℵk−2-Galois stable (3) ϕk is not categorical in any μ with μ>ℵk−2; (4) ϕk has the disjoint amalgamation property (5) For k > 2 (a) ϕk is (ℵ0, ℵk−3)-tame; indeed, syntactic first-order types determine Galois types over models of cardinality at most ℵk−3; (b) ϕk is ℵm-Galois stable for m ≤ k − 3 (c) ϕk is not (ℵk−3, ℵk−2). We adapt an example of [9]. The amalgamation, tameness, stability results, and the contrast between syntactic and Galois types are new; the categoricity results refine the earlier work of Hart and Shelah and answer a question posed by Shelah in [17].