Summary: The Degrees of Categorical Theories With Recursive Models
August 8, 2011
We show that even for categorical theories, recursiveness of the models guarantees
no information regarding the complexity of the theory. In particular, we show that ev-
ery tt-degree reducible to 0()
contains both 1-categorical and 0-categorical theories
in finite languages all of whose countable models have recursive presentations.
A fundamental question of recursive model theory is to understand the relationship
between the complexity of a theory and the complexity of presentations of its models.
It is well known that if a theory is recursive, then the Henkin construction produces a
decidable model, that is a model whose elementary diagram is recursive. Also, if T is
recursive and 1-categorical, then all of its countable models are decidably presentable
. On the other hand, if T has a recursive model, that is a model whose atomic
diagram is recursive, then in general we can only say that T is tt-reducible to 0()
Naturally, one would like to know whether this bound can be improved upon for tame
theories. Two natural classes of tame theories are the 0-categorical and 1-categorical