Indiscernibles and decidable models

Abstract

Ehrenfeucht and Mostowski [3] introduced the notion of indiscernibles and proved that every first order theory has a model with an infinite set of order indiscernibles. Since their work, techniques involving indiscernibles have proved to be extremely useful for constructing models with various specialized properties. In this paper and in a sequel [5], we investigate the effective content of Ehrenfeucht's and Mostowski's result. In this paper we consider the question of which decidable theories have decidable models with infinite recursive sets of indiscernibles. In §1, using some basic facts from stability theory, we show that certain large classes of decidable theories have decidable models with infinite recursive sets of indiscernibles. For example, we show that every ω-stable decidable theory and every stable theory which possesses a certain strong decidability property called ∃Q-decidability have such models. In §2 we construct several examples of decidable theories which have no decidable models with infinite recursive sets of indiscernibles. These examples show that our hypothesis for our positive results in §1 are necessary. Finally in §3 we give two applications of our results. First as an easy application of our results in §1, we show that every ω-stable decidable theory has uncountable models which realize only recursive types. Also our counterexamples in §2 allow us to answer negatively two questions of Baldwin and Kueker [1] concerning the effectiveness of their elimination of Ramsey quantifiers for certain theories.In [5], we show that in general the problem of finding an infinite set of indiscernibles in a decidable model is recursively equivalent to finding a path through a recursive infinite branching tree. Similarly, we show that the problem of finding an co-type of a set of indiscernibles in a decidable ω-categorical theory is recursively equivalent to finding a path through a highly recursive finitely branching tree.