When a first order \(T\) has limit models (Q2884111)

A structure \(M\) of cardinality \(\lambda\) is a \textit{superlimit}, or \textit{\(\lambda\)-superlimit}, if it is universal and for every limit ordinal \(\delta<\lambda^+\) and continuous elementary chain \((M_i:i\leq\delta)\), if \(M_i\) is isomorphic to \(M\) for all \(i<\delta\), then \(M_\delta\) is isomorphic to \(M\). This notion first occurred in the study of abstract elementary classes, but is meaningful even in the first-order context. If Th\((M)\) is superstable, then \(\lambda\)-superlimits exist iff saturated models of cardinality \(\lambda\) exist. The author exhibits a superstable complete first-order theory \(T\) such that it is consistent with ZFC that \(\aleph_1<2^{\aleph_0}\) and \(T\) has an \(\aleph_1\)-superlimit. He also considers a properly stable variant of \(T\) which has a non-saturated model of cardinality \(\aleph_1\) which is similar to being a superlimit (in fact, the author considers eight families of variants of superlimiticity.NEWLINENEWLINEFinally, the author shows that an unstable theory has no \(\lambda\)-superlimit for \(\lambda>\aleph_1+| T|\). A similar result holds for a properly stable theory, except possibly for cardinals \(\lambda<\min\{\lambda^{\aleph_0},\beth_\omega\}\).NEWLINENEWLINEAs usual, on would wish the author gave explanations rather than annotations, and tried to limit the use of symbols.