{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,31]],"date-time":"2022-03-31T11:55:05Z","timestamp":1648727705124},"reference-count":6,"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":11424,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1982,12]]},"abstract":"<jats:p>This paper is devoted to the proof of the following theorem.<\/jats:p><jats:p>Theorem. <jats:italic>Let M be a countable standard transitive model of ZF + V = L, and let \u2112 \u0404 M be a wellfounded lattice in M, with top and bottom. Let \u2223\u2112\u2223<jats:sup>M<\/jats:sup> = \u03bb, and suppose \u03ba \u2265 \u03bb is a regular cardinal in M. Then there is a generic extension N of M such that<\/jats:italic><\/jats:p><jats:p>(i) <jats:italic>N and M have the same cardinals, and <jats:sup>\u03ba<\/jats:sup>N \u2282 M;<\/jats:italic><\/jats:p><jats:p>(ii) <jats:italic>the c-degrees of sets of ordinals of N form a pattern isomorphic to \u2112;<\/jats:italic><\/jats:p><jats:p>(iii) <jats:italic>if A \u2282 On and A \u0404 N, there is B \u0404 P(\u03ba<jats:sup>+<\/jats:sup>)<jats:sup>N<\/jats:sup> such that L(A) = L(B)<\/jats:italic>.<\/jats:p><jats:p>The proof proceeds by forcing with Souslin trees, and relies heavily on techniques developed by Jech. In [5] he uses these techniques to construct simple Boolean algebras in <jats:italic>L<\/jats:italic>, and in [6] he uses them to construct a model of set theory whose <jats:italic>c<\/jats:italic>-degrees have orderlype 1 + <jats:italic>\u03c9<\/jats:italic>*.<\/jats:p><jats:p>The proof also draws on ideas of Adamovicz. In [1]\u2013[3] she obtains consistency results concerning the possible patterns of <jats:italic>c<\/jats:italic>-degrees of sets of ordinals using perfect set forcing and symmetric models. These methods have the advantage of yielding real degrees, but involve greater combinatorial complexity, in particular the use of \u2018sequential representations\u2019 of lattices.<\/jats:p><jats:p>The advantage of the approach using Souslin trees is twofold: first, we can make use of ready-made combinatorial principles which hold in <jats:italic>L<\/jats:italic>, and secondly, the notion of genericity over a Souslin tree is particularly simple.<\/jats:p>","DOI":"10.2307\/2273096","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:01:43Z","timestamp":1146938503000},"page":"739-754","source":"Crossref","is-referenced-by-count":0,"title":["Constructible lattices of <i>c<\/i>-degrees"],"prefix":"10.1017","volume":"47","author":[{"given":"C.P.","family":"Farrington","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200043644_ref006","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(75)90011-X"},{"key":"S0022481200043644_ref005","doi-asserted-by":"publisher","DOI":"10.1007\/BF02758124"},{"key":"S0022481200043644_ref002","first-page":"349","volume":"42","author":"Adamovicz","year":"1977","journal-title":"On finite lattices of degrees of constructibility"},{"key":"S0022481200043644_ref003","volume-title":"Set theory and hierarchy theory. V","author":"Adamovicz"},{"key":"S0022481200043644_ref004","volume-title":"Lecture Notes in Mathematics","author":"Devlin","year":"1973"},{"key":"S0022481200043644_ref001","first-page":"313","volume":"41","author":"Adamovicz","year":"1976","journal-title":"On finite lattices of degrees of constructibility of reals"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200043644","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,24]],"date-time":"2019-05-24T16:18:39Z","timestamp":1558714719000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200043644\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1982,12]]},"references-count":6,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1982,12]]}},"alternative-id":["S0022481200043644"],"URL":"https:\/\/doi.org\/10.2307\/2273096","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1982,12]]}}}