Abstract: By a well-known classification result in operator algebras due to George Elliott, the isomorphism class of an approximately finite C*-algebra (or simply AF-algebra) is completely determined by its dimension group. The latter is a C*-algebraic invariant which (for separable C*-algebras) takes the form of a (countable) ordered abelian group. The main result of my talk is a model theoretic version of Elliott's result in the context of infinitary logic. In particular, Elliott's arguments can be combined with a metric version of the dynamic Ehrenfeucht?Fraïssé game to show that elementary equivalence up to a rank alpha between AF-algebras is verified if elementary equivalence, up to a rank only depending on alpha, between the corresponding dimension groups holds. I will also show how this result can be used to build a class of simple AF-algebras of arbitrarily high Scott Rank.