{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,6]],"date-time":"2026-04-06T08:22:13Z","timestamp":1775463733322,"version":"3.50.1"},"reference-count":18,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":7041,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1994,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We show, roughly speaking, that it requires <jats:italic>\u03c9<\/jats:italic> iterations of the Turing jump to decode nontrivial information from Boolean algebras in an isomorphism invariant fashion. More precisely, if <jats:italic>\u03b1<\/jats:italic> is a recursive ordinal, <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200019174_inline1\"\/> is a countable structure with finite signature, and <jats:bold>d<\/jats:bold> is a degree, we say that <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200019174_inline1\"\/> has <jats:italic>\u03b1<\/jats:italic>th-<jats:italic>jump degree<\/jats:italic><jats:bold>d<\/jats:bold> if <jats:bold>d<\/jats:bold> is the least degree which is the <jats:italic>\u03b1<\/jats:italic>th jump of some degree <jats:bold>c<\/jats:bold> such there is an isomorphic copy of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200019174_inline1\"\/> with universe <jats:italic>\u03c9<\/jats:italic> in which the functions and relations have degree at most <jats:bold>c<\/jats:bold>. We show that every degree <jats:bold>d<\/jats:bold> \u2265 <jats:bold>0<\/jats:bold><jats:sup>(<jats:italic>\u03c9<\/jats:italic>)<\/jats:sup> is the <jats:italic>\u03c9<\/jats:italic>th jump degree of a Boolean algebra, but that for <jats:italic>n<\/jats:italic> &lt; <jats:italic>\u03c9<\/jats:italic> no Boolean algebra has <jats:italic>n<\/jats:italic>th-jump degree <jats:bold>d<\/jats:bold> &lt; <jats:bold>0<\/jats:bold><jats:sup>(<jats:italic>n<\/jats:italic>)<\/jats:sup>. 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.<\/jats:p>","DOI":"10.2307\/2275695","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:53:29Z","timestamp":1146941609000},"page":"1121-1138","source":"Crossref","is-referenced-by-count":18,"title":["Boolean algebras, Stone spaces, and the iterated Turing jump"],"prefix":"10.1017","volume":"59","author":[{"suffix":"Jr.","given":"Carl G.","family":"Jockusch","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Robert I.","family":"Soare","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200019174_ref001","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1990-0955487-0"},{"key":"S0022481200019174_ref017","volume-title":"Linear orderings","author":"Rosenstein","year":"1982"},{"key":"S0022481200019174_ref013","volume-title":"Regional Conference Series in Mathematics, Number 13","author":"Rabin","year":"1972"},{"key":"S0022481200019174_ref004","volume-title":"Lecture Notes in Mathematics","volume":"759","author":"Epstein","year":"1979"},{"key":"S0022481200019174_ref014","first-page":"1097","volume-title":"Handbook of Boolean Algebras","volume":"3","author":"Remmel","year":"1989"},{"key":"S0022481200019174_ref015","unstructured":"Richter L. J. , Degrees of unsolvability of models, Ph.D. thesis, University of Illinois Urbana-Champaign, Urbana, Illinois, 1977."},{"key":"S0022481200019174_ref018","unstructured":"Thurber J. , Every low2 Boolean algebra has a recursive copy (to appear)."},{"key":"S0022481200019174_ref007","volume-title":"General Topology","author":"Kelley","year":"1955"},{"key":"S0022481200019174_ref002","volume-title":"Proceeding of the American Mathematical Society","author":"Downey"},{"key":"S0022481200019174_ref005","first-page":"365","volume":"35","author":"Feiner","year":"1970","journal-title":"Hierarchies of Boolean algebras"},{"key":"S0022481200019174_ref006","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(91)90038-N"},{"key":"S0022481200019174_ref012","first-page":"1","article-title":"Decidability of second-order theories and automata on infinite trees","volume":"141","author":"Rabin","year":"1969","journal-title":"Transactions if the American Mathematical Society"},{"key":"S0022481200019174_ref016","first-page":"723","volume":"46","author":"Richter","year":"1981","journal-title":"Degrees of structures"},{"key":"S0022481200019174_ref011","first-page":"1","volume":"42","author":"Macintyre","year":"1977","journal-title":"Transfinite extensions of Friedberg's completeness criterion"},{"key":"S0022481200019174_ref008","first-page":"1034","volume":"51","author":"Knight","year":"1986","journal-title":"Degrees coded in jumps of orderings"},{"key":"S0022481200019174_ref009","first-page":"95","volume-title":"Handbook of Boolean algebras","volume":"1","author":"Koppelberg","year":"1989"},{"key":"S0022481200019174_ref003","first-page":"545","article-title":"Orderings with \u03b1th-jump degree 0(\u03b1)","volume":"114","author":"Downey","year":"1992","journal-title":"Proceedings of the American Mathematical Society"},{"key":"S0022481200019174_ref010","first-page":"189","volume-title":"Contributions to mathematical logic","author":"L\u00e4uchli","year":"1966"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200019174","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,14]],"date-time":"2019-05-14T20:40:52Z","timestamp":1557866452000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200019174\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1994,12]]},"references-count":18,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1994,12]]}},"alternative-id":["S0022481200019174"],"URL":"https:\/\/doi.org\/10.2307\/2275695","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1994,12]]}}}