The negation problem for inductive theories (Q1972818)

Let \(T\) be a first-order theory and let \(\Gamma\) be a set of closed formulas in the language of \(T\) which are not consequences of \(T\). The negation problem for the theory \(T\) and the set \(\Gamma\) is the question whether the set of formulas \(T\cup\neg(\Gamma)\) is consistent. The formula \(\neg A\) is said to be deducible by the \(CWA\)-rule if \(A\) is not a logical consequence of \(T\). Thus, the negation problem for the theory \(T\) is equivalent to the consistency problem of the \(CWA\)-rule with the theory \(T\). \textit{G. Jäger} [J. Logic Program. 8, No. 2, 229-247 (1990; Zbl 0796.68055)] showed that if the theory \(T\) is weakly countably categorical modulo some set of predicate symbols and possesses the intersection property, then the \(CWA\)-rule is consistent with \(T\). As a consequence, he proved consistency of the \(CWA\)-rule with weakly countably categorical inductive theories. In the same article, G. Jäger also stated the following problem: Do the above results hold if we consider a complete theory rather than a weakly countably categorical theory? In the article under review, a positive answer is obtained. It should be noted that, in the cited article, the term ``a weakly categorical theory'' is used instead of the term ``a weakly countably categorical theory''.