Model Theory of Rtrees
Abstract
We show the theory of pointed $\R$trees with radius at most $r$ is axiomatizable in a suitable continuous signature. We identify the model companion $\rbRT_r$ 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 builtin canonical bases for nonalgebraic types. Among the models of $\rbRT_r$ are $\R$trees that arise naturally in geometric group theory. In every infinite cardinal, we construct the maximum possible number of pairwise nonisomorphic models of $\rbRT_r$; indeed, the models we construct are pairwise nonhomeomorphic. We give detailed information about the type spaces of $\rbRT_r$. 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 $\rbRT_r$ has no atomic model.
 Publication:

arXiv eprints
 Pub Date:
 September 2018
 arXiv:
 arXiv:1810.00242
 Bibcode:
 2018arXiv181000242C
 Keywords:

 Mathematics  Logic;
 03Cxx;
 05C05 (primary);
 20F67;
 51Fxx;
 54E35;
 54F50 (secondary)
 EPrint:
 Content is the same as the published version except that a small problem in the proof of Lemma 7.7 has been fixed