Why There Are Many Nonisomorphic Models for Unsuperstable Theories

This was mentioned in [S4], and in fact in [S2]. We shall first sketch the proof and then point out some applications of the theorem and the method. We generalize the notion of indiscernibility used in Ehrenfeucht-Mostowski models (from [EM]). Let I be an (index) model, M EL model and, for each sel,äs is a (finite) sequence from M. For s = , s(l) e 7,let ä-s = äs{Q) A ••• Aäs0t_v. The indexed set {äs: sel} is called indiscernible if whenever s, t are finite sequences from I realizing the same quantifier-free type, ä-s and ä-t realize the same type in M. Now as T is not superstable, by [S2], T has formulas <p„(x, yn), a model M, and sequences äv, y e °^X such that, for 7] e <°A9 % e '% M \= <p[äv, äT] iff* % is an initial segment of rj. Clearly M has an elementary extension to a model Mi of TV By using a generalization of Ramsey's theorem [Rm] to trees (a proof was in [S3]) and by compactness, we can assume {av: TJ G7} is indiscernible; where lis a model with universe -A, one place relations Pa = X (a ^ œ), the lexicographical order < j , and the function / , f(y], z) = the lengthiest common initial segment.