Journal of Symbolic Logic

Boolean Algebras, Stone Spaces, and the Iterated Turing Jump

Carl G. Jockusch, Jr and Robert I. Soare

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We show, roughly speaking, that it requires $\omega$ iterations of the Turing jump to decode nontrivial information from Boolean algebras in an isomorphism invariant fashion. More precisely, if $\alpha$ is a recursive ordinal, $\mathscr{A}$ is a countable structure with finite signature, and $\mathbf{d}$ is a degree, we say that $\mathscr{A}$ has $\alpha$th-jump degree $\mathbf{d}$ if $\mathbf{d}$ is the least degree which is the $\alpha$th jump of some degree $\mathbf{c}$ such there is an isomorphic copy of $\mathscr{A}$ with universe $\omega$ in which the functions and relations have degree at most $\mathbf{c}$. We show that every degree $\mathbf{d} \geq 0^{(\omega)}$ is the $\omega$th jump degree of a Boolean algebra, but that for $n < \omega$ no Boolean algebra has $n$th-jump degree $\mathbf{d} > 0^{(n)}$. The former result follows easily from work of L. Feiner. The proof of the latter result uses the forcing methods of J. Knight together with an analysis of various equivalences between Boolean algebras based on a study of their Stone spaces. A byproduct of the proof is a method for constructing Stone spaces with various prescribed properties.

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J. Symbolic Logic, Volume 59, Issue 4 (1994), 1121-1138.

First available in Project Euclid: 6 July 2007

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Jockusch, Carl G.; Soare, Robert I. Boolean Algebras, Stone Spaces, and the Iterated Turing Jump. J. Symbolic Logic 59 (1994), no. 4, 1121--1138.

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