On Nonstructure of Elementary Submodels of a Stable Homogeneous Structure

We assume that M is a stable homogeneous model of large cardinality. We prove a nonstructure theorem for (slightly saturated) elementary submodels of M, assuming M has dop. We do not assume that th(M) is stable. In this paper we study elementary submodels of a stable homogeneous L-structure M. We use M as a monster model used in “classical” stability theory and so as in [HS], we assume that |M| is strongly inaccessible (= regular and strong limit) and > |L|, where |L| is the number of L-formulas. We recall that by [Sh1], if D is a stable finite diagram, then it has a monster model like M (assuming, of course, the existence of a strongly inaccessible cardinal). As in [HS], we can drop the assumption of |M| being strongly inaccessible if instead of all elementary submodels of M, we study only suitably small ones. We assume that the reader is familiar with [HS] and we use conventions, notions and results of [HS] freely. The machinery in [HS] is an improved version of that in [Hy1]. Now we modify it so that it comes closer to the lines of [Hy1]. As mentioned in the abstract, we prove a nonstructure theorem for (FM λr(M)-saturated) elementary submodels of M, assuming M has dop (= λr(M)-dop). By a nonstructure theorem we mean a theorem which implies, at least, that for most κ, the number of models of power κ is the maximal one. Often nonstructure theorems imply also that a “Shelah-style” structure theorem does not hold for a class of models. See [HT] for further discussion about nonstructure theorems. In [HS] a structure theorem was proved for FM λr(M)-saturated models assuming M is superstable and does not have λr(M)-dop. So in case M is superstable, we have a dichotomy for 1991 Mathematics Subject Classification: 03C45, 03C52.