ABSTRACT
In this part of the book we study S∞-orbit equivalence relations. In Sections 2.4 and 3.6 we have already seen connections between S∞-orbit equivalence relations and isomorphism relations for countable models. In this part we will establish a full duality between the two subjects. Historically many of the model theoretic results predated and in fact motivated their group action counterpart. On the other hand, the study of isomorphism relation of countable structures also benefited from the perspective of Polish group actions. Since the realization that they are the same subject our understanding of countable model theory has advanced significantly.
