We consider the Turing degrees of prime models of complete decidable
theories. In particular we show that every complete decidable atomic
theory has a prime model whose elementary diagram is low. We combine
the construction used in the proof with other constructions to show
that complete decidable atomic theories have low prime models with added
properties.
¶
If we have a complete decidable atomic theory with all types of the
theory computable, we show that for every degree d with
0 < d ≤ 0’,
there is a prime model with elementary diagram of degree d. Indeed,
this is a corollary of the fact that if T is a complete decidable theory
and L is a
computable set of c.e. partial types of T, then for any
Δ02 degree d > 0,
T has a d-decidable model omitting the
nonprincipal types listed by L.