{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,10,23]],"date-time":"2023-10-23T02:41:28Z","timestamp":1698028888911},"reference-count":7,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":11150,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1983,9]]},"abstract":"<jats:p>The <jats:italic>Sikorski Extension Theorem<\/jats:italic> [6] states that, for any Boolean algebra <jats:italic>A<\/jats:italic> and any complete Boolean algebra <jats:italic>B<\/jats:italic>, any homomorphism of a subalgebra of <jats:italic>A<\/jats:italic> into <jats:italic>B<\/jats:italic> can be extended to the whole of <jats:italic>A<\/jats:italic>. That is,<\/jats:p><jats:p><jats:bold>Inj<\/jats:bold>: <jats:italic>Any complete Boolean algebra is injective<\/jats:italic> (in the category of Boolean algebras).<\/jats:p><jats:p>The proof of <jats:bold>Inj<\/jats:bold> uses the axiom of choice (<jats:bold>AC<\/jats:bold>); thus the implication <jats:bold>AC<\/jats:bold> \u2192 <jats:bold>Inj<\/jats:bold> can be proved in Zermelo-Fraenkel set theory (<jats:bold>ZF<\/jats:bold>). On the other hand, the <jats:italic>Boolean prime ideal theorem<\/jats:italic><\/jats:p><jats:p><jats:bold>BPI<\/jats:bold>: <jats:italic>Every Boolean algebra contains a prime ideal (or, equivalently, an ultrafilter)<\/jats:italic><\/jats:p><jats:p>may be equivalently stated as:<\/jats:p><jats:p><jats:italic>The two element Boolean algebra 2 is injective<\/jats:italic>,<\/jats:p><jats:p>and so the implication <jats:bold>Inj<\/jats:bold> \u2192 <jats:bold>BPI<\/jats:bold> can be proved in <jats:bold>ZF<\/jats:bold>.<\/jats:p><jats:p>In [3], Luxemburg surmises that this last implication cannot be reversed in <jats:bold>ZF<\/jats:bold>. It is the main purpose of this paper to show that this surmise is correct. We shall do this by showing that <jats:bold>Inj<\/jats:bold> implies that <jats:bold>BPI<\/jats:bold> holds in every Boolean extension of the universe of sets, and then invoking a recent result of Monro [5] to the effect that <jats:bold>BPI<\/jats:bold> does <jats:italic>not<\/jats:italic> yield this conclusion.<\/jats:p>","DOI":"10.2307\/2273477","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:04:45Z","timestamp":1146938685000},"page":"841-846","source":"Crossref","is-referenced-by-count":6,"title":["On the strength of the Sikorski extension theorem for Boolean algebras"],"prefix":"10.1017","volume":"48","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":"S0022481200038020_ref004","volume-title":"Lectures on Boolean-valued models for set theory","author":"Scott","year":"1967"},{"key":"S0022481200038020_ref002","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0061160"},{"key":"S0022481200038020_ref001","volume-title":"Boolean-valued models and Independence proofs in set theory","author":"Bell","year":"1977"},{"key":"S0022481200038020_ref005","first-page":"39","volume":"48","author":"Monro","year":"1983","journal-title":"On generic extensions without the axiom of choice"},{"key":"S0022481200038020_ref006","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 Polonaise de Math\u00e9matique"},{"key":"S0022481200038020_ref007","doi-asserted-by":"publisher","DOI":"10.2307\/1970860"},{"key":"S0022481200038020_ref003","doi-asserted-by":"publisher","DOI":"10.4064\/fm-55-3-239-247"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200038020","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,23]],"date-time":"2019-05-23T18:37:01Z","timestamp":1558636621000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200038020\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1983,9]]},"references-count":7,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1983,9]]}},"alternative-id":["S0022481200038020"],"URL":"https:\/\/doi.org\/10.2307\/2273477","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1983,9]]}}}