{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,30]],"date-time":"2022-03-30T11:30:34Z","timestamp":1648639834298},"reference-count":13,"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":9598,"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1987,12]]},"abstract":"<jats:p>The Turing jump can easily be iterated any finite number of times. The challenge is posed by by transfinite iterations. <jats:italic>a<\/jats:italic><jats:sup>(<jats:italic>\u03c9<\/jats:italic>)<\/jats:sup> has an intuitively compelling definition: {\u2039<jats:italic>m, n<\/jats:italic>\u203a: <jats:italic>n<\/jats:italic> \u2208 <jats:italic>a<\/jats:italic><jats:sup>(<jats:italic>m<\/jats:italic>)<\/jats:sup>}. Starting with Putnam et al. [1], [2], [9] and culminating in Jockusch-Simpson [8] and Hodes [6], recent work has justified this <jats:italic>\u03c9<\/jats:italic>-jump and extended it through <jats:inline-graphic xmlns:xlink=\"https:\/\/2.zoppoz.workers.dev:443\/http\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200029340_inline1\" \/>.<\/jats:p><jats:p>Of great use are the master codes of Jensen [3], [7]. Briefly, a <jats:italic>\u0394<jats:sub>n<\/jats:sub><\/jats:italic>(<jats:italic>L<jats:sub>\u03b2<\/jats:sub><\/jats:italic>)-master code is a complete <jats:italic>\u0394<jats:sub>n<\/jats:sub><\/jats:italic>(<jats:italic>L<jats:sub>\u03b2<\/jats:sub><\/jats:italic>) set of ordinals, with supremum as small as possible. Connections with classical recursion theory are tight. When the <jats:italic>\u0394<jats:sub>n<\/jats:sub><\/jats:italic>(<jats:italic>L<jats:sub>\u03b2<\/jats:sub><\/jats:italic>) master codes are reals, they are Turing equivalent. If <jats:italic>a<\/jats:italic> is a <jats:italic>\u0394<jats:sub>n<\/jats:sub><\/jats:italic>(<jats:italic>L<jats:sub>\u03b2<\/jats:sub><\/jats:italic>) MC, then <jats:italic>a\u2032<\/jats:italic> is a <jats:italic>\u0394<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic> + 1<\/jats:sub>(<jats:italic>L<jats:sub>\u03b2<\/jats:sub><\/jats:italic>) MC. Intuitively satisfying transfinite jumps, such as the <jats:italic>\u03c9<\/jats:italic>-jump above, produce a Turing jump hierarchy equal to an initial segment of the master codes.<\/jats:p><jats:p>Hodes capitalized on these facts by defining 0<jats:sup><jats:italic>\u03b1<\/jats:italic><\/jats:sup>, <jats:italic>\u03b1<\/jats:italic> &lt; <jats:inline-graphic xmlns:xlink=\"https:\/\/2.zoppoz.workers.dev:443\/http\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200029340_inline2\" \/>, set-theoretically as the degree of the <jats:italic>\u03b1<\/jats:italic>th MC which is a real, and then proving the equivalence of a jump-theoretic definition of 0<jats:sup><jats:italic>\u03b1<\/jats:italic><\/jats:sup>. The successor codes are jumps of their predecessors, and for a limit <jats:italic>\u03bb<\/jats:italic>, 0<jats:sup><jats:italic>\u03bb<\/jats:italic><\/jats:sup> is the minimum of a set of degrees associated with {0<jats:sup><jats:italic>\u03b1<\/jats:italic><\/jats:sup>: <jats:italic>\u03b1<\/jats:italic>: &lt; <jats:italic>\u03bb<\/jats:italic>}.<\/jats:p>","DOI":"10.2307\/2273829","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:23:41Z","timestamp":1146954221000},"page":"952-958","source":"Crossref","is-referenced-by-count":1,"title":["Uncountable master codes and the jump hierarchy"],"prefix":"10.1017","volume":"52","author":[{"given":"Robert S.","family":"Lubarsky","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200029340_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-11035-5"},{"key":"S0022481200029340_ref003","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1969-0242673-X"},{"key":"S0022481200029340_ref013","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(78)90026-8"},{"key":"S0022481200029340_ref002","first-page":"497","volume":"33","author":"Boolos","year":"1968","journal-title":"Degrees of unsolvability of constructible sets of integers"},{"key":"S0022481200029340_ref006","doi-asserted-by":"publisher","DOI":"10.1007\/BF02761905"},{"key":"S0022481200029340_ref007","first-page":"204","volume":"45","author":"Hodes","year":"1980","journal-title":"Jumping through the transfinite"},{"key":"S0022481200029340_ref008","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(72)90001-0"},{"key":"S0022481200029340_ref009","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(76)90023-1"},{"key":"S0022481200029340_ref010","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(08)71236-9"},{"key":"S0022481200029340_ref011","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(74)90005-9"},{"key":"S0022481200029340_ref012","doi-asserted-by":"publisher","DOI":"10.1090\/pspum\/013.1\/0276079"},{"key":"S0022481200029340_ref005","doi-asserted-by":"publisher","DOI":"10.2307\/1971132"},{"key":"S0022481200029340_ref004","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0059290"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200029340","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,20]],"date-time":"2019-05-20T21:17:12Z","timestamp":1558387032000},"score":1,"resource":{"primary":{"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/www.cambridge.org\/core\/product\/identifier\/S0022481200029340\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1987,12]]},"references-count":13,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1987,12]]}},"alternative-id":["S0022481200029340"],"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/doi.org\/10.2307\/2273829","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1987,12]]}}}