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Boolean algebras, Stone spaces, and the iterated Turing jump. (English) Zbl 0819.03034
Let \(B\) be a model. The degree of \(B\), \(\deg (B)\) is the degree of the join of functions and relations in \(B\). Following C. J. Ash, the first author and J. F. Knight [Trans. Am. Math. Soc. 319, 573-599 (1990; Zbl 0705.03022)], the \(\alpha\)-th jump degree of \(B\) is the least degree, if any, of the form \((\deg (A))^{(\alpha)}\) for \(A\) isomorphic to \(B\). The authors analyse the possible \(\alpha\)-jump degrees of countable Boolean algebras. They show that for \({\mathbf a}\geq \mathbf{0}^{(\omega)}\) there is a Boolean algebra with \(\omega\)-jump degree for \(n<\omega\). (This is called a proper \(\omega\)-jump degree). On the other hand, they show that for \(n<\omega\), if a Boolean algebra has \(n\)-th jump degree then \(n= \mathbf{0}^{({\mathbf n})}\). This latter result uses a forcing argument plus some pretty algebra. Downey-Jockusch and Thurber have proven that there is no Boolean algebra with proper 1- or 2-jump degrees \(\mathbf{0}{\mathbf '}\) and \(\mathbf{0}{\mathbf {''}}\) respectively. It is an open question of whether it is possible to have a Boolean algebra with proper \(n\)-jump degree for any \(n<\omega\). Also open is the question of whether a Boolean algebra of finite Cantor-Bendixson rank can be arithmetical yet not isomorphic to a recursive Boolean algebra.

MSC:
03D45 Theory of numerations, effectively presented structures
03D25 Recursively (computably) enumerable sets and degrees
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