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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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References:
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