An inelastic model with indiscernibles

Abstract

Let L be a countable language including the unary relation symbol U. Let and be L-structures such that is a proper elementary U-extension of ; i.e., , and . Under what conditions will have a proper elementary U-extension? In [2], it was shown that this is not always the case, even if and are countable. However, the examples given are completely artificial, and it still seems that in most cases will have a proper elementary U-extension.Lascar asked whether will necessarily have a proper elementary U-extension whenever it contains an infinite set of indiscernibles over . This paper gives a counterexample for Lascar's question. The example is produced by modifying one of the examples in [2], using an idea of Marcus [5].Models containing an infinite set of indiscernibles can often be “stretched” to produce larger models that share some desired nonelementary property with the original [1], [6]. However, the mere presence of indiscernibles in a model does not guarantee that it can be used in this way.If the model is not completely determined by the indiscernibles, the nonelementary property may not carry over to larger models. An example of this is given in [3]. The example for Lascar's question is further evidence that models with indiscernibles need not be “elastic”.