Constructing \(\omega\)-stable structures: Model completeness.
The authors extend work by \textit{K. Holland} [J. Symb. Log. 64, No. 3, 946--962 (1999; Zbl 0945.03045)] by showing that the projective plane of \textit{J. T. Baldwin} [Trans. Am. Math. Soc. 342, No. 2, 695--711 (1994; Zbl 0796.03041)] and the finite rank bicoloured fields of \textit{J. T. Baldwin} and \textit{K. Holland} [J. Symb. Log. 65, No. 1, 371--391 (2000; Zbl 0957.03044)] are model-complete, while the infinite rank bicoloured field of \textit{B. Poizat} [J. Symb. Log. 64, No. 3, 1339--1355 (1999; Zbl 0938.03058)] is not. More generally, the finite rank collaps of a strongly minimal set expanded by a random unary predicate is almost strongly minimal and model-complete provided the strongly minimal set has a dense set of exactly rank \(k\) formulas, where \(k\) is the rank of the collapse (and this excludes triviality for \(k>2\)). The proof in the second case makes use of Lindström's theorem that inductive theories categorical in an infinite power are model-complete; in the first case it uses a new (but old-style) characterization of almost strongly minimal model-complete theories.
scientific article; zbMATH DE number 2056209
scientific article