Degrees of indiscernibles in decidable models

Abstract

ABSTRACT. We show that the problem of finding an infinite set of indiscernibles in an arbitrary decidable model of a first order theory is essentially equivalent to the problem of finding an infinite path through a recursive wbranching tree. Similarly, we show that the problem of finding an infinite set of indiscernibles in a decidable model of an u>-categorical theory with decidable atoms is essentially equivalent to finding an infinite path through a recursive binary tree. Introduction.Ehrenfeucht and Mostowski [2] introduced the notion of indiscernibles and proved that every first order theory has a model with an infinite set of order indiscernibles. In [6], we investigated the question of which decidable theories have decidable models with infinite recursive sets of indiscernibles.For example, we showed that every w-stable decidable theory and every stable theory which has a certain strong decidability property called BQ-decidability have such models. Moreover, we gave a series of examples of decidable theories which have no decidable models with infinite recursive sets of indiscernibles which show that the various hypotheses of our positive results are necessary.In this paper, we investigate the possible degrees of indiscernibles in a decidable model Ai. We shall show, in a sense to be made precise in §1, that the problem of finding a set of indiscernibles in an arbitrary decidable model M of a first order theory T is essentially equivalent to the problem of finding an infinite path through a recursive w-branching tree T. Similarly, we shall show that the problem of finding an infinite set of indiscernibles in a decidable model Ai of an w-categorical theory T with decidable atoms is essentially equivalent to finding an infinite path through