The spectrum problem. III: Universal theories
[For Parts I and II see ibid. 43, 324-356, 357-364 (1982; Zbl 0532.03013 and Zbl 0532.03014.] It is proved that for stable complete T and C a big model of T, either A) for all elementary submodels \(M_ 0<M_ 1,M_ 2\) of C with \(M_ 1\) and \(M_ 2\) independent over \(M_ 0\), the substructure of C generated by \(M_ 1\cup M_ 2\) is an elementary substructure, or B) the universal part of T has an unstable completion. By Shelah theory it is concluded that if T is a universal (not necessarily complete) theory then either there is a structure theorem for the models of T (specifically any model is generated by a nonforking tree of small models) or T has \(2^{\lambda}\) models in every cardinality \(\lambda >| T|\). Examples are also given of, for all \(n<\omega\), a universal theory \(T_ n\) which is unstable but whose instability is not witnessed by a \(\Sigma_ n\)-formula.
scientific article; zbMATH DE number 4029522