Computable Boolean algebras (Q2710598)

Feiner first proved that there are \(\Delta^0_2\) Boolean algebras that do not have computable copies. Feiner's construction actually proves that there are \(\Delta^0_2\) Boolean algebras that have no copy which is \(\text{low}_n\) for \(n\in\omega\). In contrast with the situation for linear orderings, \textit{R. Downey} and \textit{C. G. Jockusch} [Proc. Am. Math. Soc. 122, 871-880 (1994; Zbl 0820.03019)] showed that every low Boolean algebra has a computable copy, and this result was later extended to \(\text{low}_2\) by \textit{J. J. Thurber} [Proc. Am. Math. Soc. 123, 3859-3866 (1995; Zbl 0840.03024)]. The conjecture is that every \(\text{low}_n\) Boolean algebra is isomorphic to a computable one. The authors verify this conjecture for \(n\leq 4\). Whilst this might seem scant progress, in fact the proof is significantly more complex than either of the predecessors and crucially depends on a strong generalization of the Remmel-Vaught theorem on the atom ideal. This paper makes the general question look even more difficult than first supposed.