{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,5]],"date-time":"2025-10-05T04:36:53Z","timestamp":1759639013939},"reference-count":10,"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":5945,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1997,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We analyze Zorn's Lemma and some of its consequences for Boolean algebras in a constructive setting. We show that Zorn's Lemma is persistent in the sense that, if it holds in the underlying set theory, in a properly stated form it continues to hold in all intuitionistic type theories of a certain natural kind. (Observe that the axiom of choice cannot be persistent in this sense since it implies the law of excluded middle.) We also establish the persistence of some familiar results in the theory of (complete) Boolean algebras\u2014notably, the proposition that every complete Boolean algebra is an absolute subretract. This (almost) resolves a question of Banaschewski and Bhutani as to whether the Sikorski extension theorem for Boolean algebras is persistent.<\/jats:p>","DOI":"10.2307\/2275642","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T23:03:06Z","timestamp":1146956586000},"page":"1265-1279","source":"Crossref","is-referenced-by-count":10,"title":["Zorn's lemma and complete Boolean algebras in intuitionistic type theories"],"prefix":"10.1017","volume":"62","author":[{"given":"J. L.","family":"Bell","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200015747_ref010","first-page":"332","article-title":"A theorem on extensions of homomorphisms","volume":"21","author":"Sikorski","year":"1948","journal-title":"Annales de la Soci\u00e9t\u00e9 Pol. de Math\u00e9matiques"},{"key":"S0022481200015747_ref008","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0061829"},{"key":"S0022481200015747_ref007","volume-title":"Lectures on Boolean algebras","author":"Halmos","year":"1963"},{"key":"S0022481200015747_ref009","first-page":"39","volume":"48","author":"Monro","year":"1983","journal-title":"On generic extensions without the axiom of choice"},{"key":"S0022481200015747_ref004","first-page":"841","volume":"48","author":"Bell","year":"1983","journal-title":"On the strength of the Sikorski extension theorem for Boolean algebras"},{"key":"S0022481200015747_ref006","volume-title":"Toposes and local set theories: An introduction","author":"Bell","year":"1988"},{"key":"S0022481200015747_ref002","first-page":"501","article-title":"On the injectivity of Boolean algebras","volume":"34","author":"Banaschewski","year":"1993","journal-title":"Commentationes Mathematiae Universitatis Carolinae"},{"key":"S0022481200015747_ref001","first-page":"1","article-title":"Compact regular frames and the Sikorski theorem","volume":"28","author":"Banaschewski","year":"1988","journal-title":"Kyungpook Mathematical Journal"},{"key":"S0022481200015747_ref005","doi-asserted-by":"publisher","DOI":"10.4064\/fm-130-1-51-55"},{"key":"S0022481200015747_ref003","doi-asserted-by":"publisher","DOI":"10.1017\/S0305004100065853"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200015747","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,11]],"date-time":"2019-05-11T19:55:23Z","timestamp":1557604523000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200015747\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1997,12]]},"references-count":10,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1997,12]]}},"alternative-id":["S0022481200015747"],"URL":"https:\/\/doi.org\/10.2307\/2275642","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1997,12]]}}}