Model completeness for trivial, uncountably categorical theories of Morley rank \(1\) (Q862348)

The main theorem: If \(T\) is a trivial uncountably categorical theory of Morley rank 1, then the theory obtained from \(T\) by naming constants for a model is model complete. This result is in some senses optimal. There exist trivial Morley rank 1 theories which are not categorical and for which the conclusion of the theorem fails. Also, \textit{D. Marker} [``Non \(\Sigma_n\) axiomatizable almost strongly minimal theories'', J. Symb. Log. 54, 921--927 (1989; Zbl 0698.03021)] constructed trivial totally categorical theories of Morley rank 2 which are not model complete after naming any set of constants.