Description:
It is shown that every (countable) Boolean algebra with a presentation of low Turing degree is isomorphic to a recursive Boolean algebra. This contrasts with a result of Feiner (1967) that there is a Boolean algebra with a presentation of degree ≤ 0 ′ \leq 0’ which is not isomorphic to a recursive Boolean algebra. It is also shown that for each n there is a finitely axiomatizable theory T n {T_n} such that every low n {\text {low}_n} model of T n {T_n} is isomorphic to a recursive structure but there is a low n + 1 {\text {low}_{n + 1}} model of T n {T_n} which is not isomorphic to any recursive structure. In addition, we show that n + 2 n + 2 is the Turing ordinal of the same theory T n {T_n} , where, very roughly, the Turing ordinal of a theory describes the number of jumps needed to recover nontrivial information from models of the theory. These are the first known examples of theories with Turing ordinal α \alpha for 3 ≤ α > ω 3 \leq \alpha > \omega .