EVERY LOW BOOLEAN-ALGEBRA IS ISOMORPHIC TO A RECURSIVE ONE

It is shown that every (countable) Boolean algebra with a presentation of low Turing degree is isomorphic to a recursive Boolean algebra. Th is contrasts with a result of Feiner (1967) that there is a Boolean al gebra with a presentation of degree less than or equal to 0' which is not isomorphic to a recursive Boolean algebra. It is also shown that f or each n there is a finitely axiomatizable theory T-n such that every low(n) model of T-n is isomorphic to a recursive structure but there is a low(n+I) model of T-n which is not isomorphic to any recursive st ructure. In addition, we show that n + 2 is the Turing ordinal of the same theory T-n, where, very roughly, the Turing ordinal of a theory d escribes the number of jumps needed to recover nontrivial information from models of the theory. These are the first known examples of theor ies with Turing ordinal alpha for 3 less than or equal to alpha < omeg a.