Boolean Algebras, Stone Spaces, and the Iterated Turing Jump

Journal of Symbolic Logic 59 (4):1121 - 1138 (1994)
Abstract
We show, roughly speaking, that it requires ω iterations of the Turing jump to decode nontrivial information from Boolean algebras in an isomorphism invariant fashion. More precisely, if α is a recursive ordinal, A is a countable structure with finite signature, and d is a degree, we say that A has αth-jump degree d if d is the least degree which is the αth jump of some degree c such there is an isomorphic copy of A with universe ω in which the functions and relations have degree at most c. We show that every degree d ≥ 0 (ω) is the ωth jump degree of a Boolean algebra, but that for $n no Boolean algebra has nth-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
Keywords No keywords specified (fix it)
Categories (categorize this paper)
DOI 10.2307/2275695
Options
 Save to my reading list
Follow the author(s)
Edit this record
My bibliography
Export citation
Find it on Scholar
Mark as duplicate
Request removal from index
Revision history
Download options
Our Archive


Upload a copy of this paper     Check publisher's policy     Papers currently archived: 29,848
Through your library
References found in this work BETA

Add more references

Citations of this work BETA
Measuring Complexities of Classes of Structures.Barbara F. Csima & Carolyn Knoll - 2015 - Annals of Pure and Applied Logic 166 (12):1365-1381.

Add more citations

Similar books and articles
Added to PP index
2009-01-28

Total downloads
24 ( #230,461 of 2,210,508 )

Recent downloads (6 months)
1 ( #389,893 of 2,210,508 )

How can I increase my downloads?

Monthly downloads
My notes
Sign in to use this feature