Model theory of R-trees

We show the theory of pointed -trees with radius at most r is axiomatizable in a suitable continuous signature. We identify the model companion ᵣ of this theory and study its properties. In particular, the model companion is complete and has quantifier elimination; it is stable but not superstable. We identify its independence relation and find built-in canonical bases for non-algebraic types. Among the models of ᵣ are -trees that arise naturally in geometric group theory. In every infinite cardinal, we construct the maximum possible number of pairwise non-isomorphic models of ᵣ; indeed, the models we construct are pairwise non-homeomorphic. We give detailed information about the type spaces of ᵣ. Among other things, we show that the space of $2$-types over the empty set is nonseparable. Also, we characterize the principal types of finite tuples (over the empty set) and use this information to conclude that ᵣ has no atomic model.