{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,6]],"date-time":"2026-04-06T22:09:42Z","timestamp":1775513382297,"version":"3.50.1"},"reference-count":6,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":12429,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1980,3]]},"abstract":"<jats:p>Consider the following propositions:<\/jats:p><jats:p>(A) Every uncountable subset of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200047174_inline1\"\/> contains an uncountable chain or antichain (with respect to \u2286).<\/jats:p><jats:p>(B) Every uncountable Boolean algebra contains an uncountable antichain (i.e., an uncountable set of pairwise <jats:italic>incomparable<\/jats:italic> elements).<\/jats:p><jats:p>Until quite recently, relatively little was known about these propositions. The oldest result, due to Kunen [4] and the author independently, asserts that if the Continuum Hypothesis (CH) holds, then (A) is false. In fact there is a counter-example \u3008<jats:italic>A<jats:sub>\u03b1<\/jats:sub><\/jats:italic>: <jats:italic>\u03b1<\/jats:italic> &lt; <jats:italic>\u03c9<\/jats:italic><jats:sub>1<\/jats:sub>\u3009 such that <jats:italic>\u03b1<\/jats:italic> &lt; <jats:italic>\u03b2<\/jats:italic> implies <jats:italic>A<jats:sub>\u03b2<\/jats:sub> \u2212<\/jats:italic><jats:italic>A<jats:sub>\u03b1<\/jats:sub><\/jats:italic> is finite. Kunen also observed that Martin's Axiom (MA) + \u00acCH implies that no such counterexample \u3008<jats:italic>A<jats:sub>\u03b1<\/jats:sub><\/jats:italic>: <jats:italic>\u03b1<\/jats:italic> &lt; <jats:italic>\u03c9<\/jats:italic><jats:sub>1<\/jats:sub>\u3009 exists.<\/jats:p><jats:p>Much later, Komj\u00e1th and the author [2] showed that \u25ca implies the existence of several kinds of uncountable Boolean algebras with no uncountable chains or antichains. Similar results (but motivated quite differently) were obtained independently by Rubin [5]. Berney [3] showed that CH implies that (B) is false, but his algebra has uncountable chains. Finally, Shelah showed very recently that CH implies the existence of an uncountable Boolean algebra with no uncountable chains or antichains.<\/jats:p><jats:p>Except for Kunen's result cited above, the only result in the other direction was the theorem, due also to Kunen, that MA + \u00acCH implies that any uncountable subset of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200047174_inline1\"\/> with no uncountable antichains must have both ascending and decending infinite sequences under \u2286.<\/jats:p>","DOI":"10.2307\/2273356","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:51:56Z","timestamp":1146952316000},"page":"85-92","source":"Crossref","is-referenced-by-count":16,"title":["Chains and antichains in"],"prefix":"10.1017","volume":"45","author":[{"given":"James E.","family":"Baumgartner","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200047174_ref006","doi-asserted-by":"publisher","DOI":"10.2307\/1970860"},{"key":"S0022481200047174_ref005","unstructured":"Rubin M. , A Boolean algebra with few subalgebras, interval Boolean algebras and retractiveness (to appear)."},{"key":"S0022481200047174_ref001","doi-asserted-by":"publisher","DOI":"10.4064\/fm-79-2-101-106"},{"key":"S0022481200047174_ref002","volume-title":"Fundamenta Mathematicae","author":"Baumgartner"},{"key":"S0022481200047174_ref003","unstructured":"Berney E. S. and Nyikos P. J. , The length and breadth of Boolean algebras (to appear)."},{"key":"S0022481200047174_ref004","unstructured":"Kunen K. , mimeographed notes."}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200047174","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,26]],"date-time":"2019-05-26T19:36:48Z","timestamp":1558899408000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200047174\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1980,3]]},"references-count":6,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1980,3]]}},"alternative-id":["S0022481200047174"],"URL":"https:\/\/doi.org\/10.2307\/2273356","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1980,3]]}}}