{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,29]],"date-time":"2025-09-29T11:52:29Z","timestamp":1759146749484},"reference-count":11,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":1745,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2009,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Hirschfeldt and Shore have introduced a notion of stability for infinite posets. We define an arguably more natural notion called weak stability, and we study the existence of infinite computable or low chains or antichains, and of infinite <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200003650_inline1\" \/> chains and antichains, in infinite computable stable and weakly stable posets. For example, we extend a result of Hirschfeldt and Shore to show that every infinite computable weakly stable poset contains either an infinite low chain or an infinite computable antichain. Our hardest result is that there is an infinite computable weakly stable poset with no infinite <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200003650_inline1\" \/> chains or antichains. On the other hand, it is easily seen that every infinite computable stable poset contains an infinite computable chain or an infinite <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200003650_inline1\" \/> antichain. In Reverse Mathematics, we show that SCAC, the principle that every infinite stable poset contains an infinite chain or antichain, is equivalent over RCA<jats:sub>0<\/jats:sub> to WSAC, the corresponding principle for weakly stable posets.<\/jats:p>","DOI":"10.2178\/jsl\/1243948336","type":"journal-article","created":{"date-parts":[[2009,6,2]],"date-time":"2009-06-02T13:12:25Z","timestamp":1243948345000},"page":"693-711","source":"Crossref","is-referenced-by-count":3,"title":["Stability and posets"],"prefix":"10.1017","volume":"74","author":[{"suffix":"Jr.","given":"Carl G.","family":"Jockusch","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Bart","family":"Kastermans","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Steffen","family":"Lempp","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Manuel","family":"Lerman","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Reed","family":"Solomon","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200003650_ref008","first-page":"489","volume":"64","author":"Hummel","year":"1999","journal-title":"Generalized cohesiveness"},{"key":"S0022481200003650_ref007","first-page":"1301","volume":"59","author":"Hummel","year":"1994","journal-title":"Effective versions of Ramsey's theorem: Avoiding the cone above 0\u2032"},{"key":"S0022481200003650_ref006","first-page":"171","volume":"72","author":"Hirschfeldt","year":"2007","journal-title":">Combinatorial principles weaker than Ramsey's theorem for pairs"},{"key":"S0022481200003650_ref004","first-page":"923","volume":"66","author":"Herrmann","year":"2001","journal-title":"Infinite chains and antichains in computable partial orderings"},{"key":"S0022481200003650_ref001","first-page":"1","volume":"66","author":"Cholak","year":"2001","journal-title":"On the strength of Ramsey's theorem for pairs"},{"key":"S0022481200003650_ref010","volume-title":"Theory of Recursive Functions and Effective Computability","author":"Rogers","year":"1967"},{"key":"S0022481200003650_ref009","first-page":"268","volume":"37","author":"Jockusch","year":"1972","journal-title":"Ramsey's theorem and recursion theory"},{"key":"S0022481200003650_ref002","first-page":"85","article-title":"Remarks on l-genericity, semigenericity, and related concepts","volume":"28","author":"Demuth","year":"1987","journal-title":"Commentationes Mathematicae Universitatis Carolinae"},{"key":"S0022481200003650_ref011","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-59971-2"},{"key":"S0022481200003650_ref005","first-page":"143","volume-title":"Proceedings of the Program on Computational Prospects of Infinity","author":"Hirschfeldt","year":"2007"},{"key":"S0022481200003650_ref003","doi-asserted-by":"publisher","DOI":"10.1007\/s00153-008-0114-2"}],"container-title":["The Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200003650","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,30]],"date-time":"2019-04-30T13:26:21Z","timestamp":1556630781000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200003650\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,6]]},"references-count":11,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2009,6]]}},"alternative-id":["S0022481200003650"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1243948336","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2009,6]]}}}