## Journal of Symbolic Logic

### Boolean Algebras, Stone Spaces, and the Iterated Turing Jump

#### Abstract

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.

#### Article information

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

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183744615

Mathematical Reviews number (MathSciNet)
MR1312300

Zentralblatt MATH identifier
0819.03034

JSTOR