A test for expandability (Q1128187)

Earlier the author [J. Symb. Log. 60, No. 2, 673-683 (1995; Zbl 0827.03021)] introduced the notion of a compactly expandable model. The notion of expandability was implicitly introduced by \textit{S. Shelah} [Ann. Math. Logic 4, 75-114 (1972; Zbl 0243.02039)] and \textit{M. Morley} [``Countable models with standard part'', in: P. Suppes et al. (eds.), Logic, Methodology and Philosophy of Science. IV, Stud. Log. Found. Math. 74, 57-62 (1973; Zbl 0487.03002)]. It is unknown whether every compactly expandable model is expandable or not. In the paper under review, a test for expandability is given. It is used to prove that a compactly expandable model of cardinality \(\geq 2^{\omega}\) of a superstable theory is expandable.