{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,2]],"date-time":"2022-04-02T11:10:10Z","timestamp":1648897810262},"reference-count":20,"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":6859,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1995,6]]},"abstract":"<jats:p>Since Georg Cantor discovered set theory the main problem in this area of mathematical research has been to discover what is the size of the continuum. The continuum hypothesis (<jats:bold>CH<\/jats:bold>) says that every infinite set of reals either has the same cardinality as the set of all reals or has the cardinality of the set of natural numbers, namely<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200018521_eqnU1\" \/><\/jats:disp-formula><\/jats:p><jats:p>In 1939 Kurt G\u00f6del discovered the Constructible Universe and proved that <jats:bold>CH<\/jats:bold> holds in it. In the early sixties Paul Cohen proved that every universe of set theory can be extended to a bigger universe of set theory where <jats:bold>CH<\/jats:bold> fails. Moreover, given any reasonable cardinal <jats:italic>\u03ba<\/jats:italic>, it is possible to build a model where the continuum size is <jats:italic>\u03ba<\/jats:italic>. The new technique discovered by Cohen is called <jats:italic>forcing<\/jats:italic> and is being used successfully in other branches of mathematics (analysis, algebra, graph theory, etc.).<\/jats:p><jats:p>In the light of these two stupendous works the experts (especially the platonists) were forced to conclude that from the point of view of the classical axiomatization of set theory (called <jats:bold>ZFC<\/jats:bold>) it is impossible to give any answer to the continuum size problem: <jats:italic>everything is possible!<\/jats:italic><\/jats:p><jats:p>In private communications G\u00f6del suggested that the continuum size from a platonistic point of view should be <jats:italic>\u03c9<\/jats:italic><jats:sub>2<\/jats:sub>, the second uncountable cardinal. As this is not provable in <jats:bold>ZFC<\/jats:bold>, G\u00f6del suggested that a new axiom should be added to <jats:bold>ZFC<\/jats:bold> to decide that the cardinality of the continuum is <jats:italic>\u03c9<\/jats:italic><jats:sub>2<\/jats:sub>.<\/jats:p>","DOI":"10.2307\/2275837","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:55:05Z","timestamp":1146956105000},"page":"374-391","source":"Crossref","is-referenced-by-count":1,"title":["Martin's axiom and the continuum"],"prefix":"10.1017","volume":"60","author":[{"given":"Haim","family":"Judah","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Andrzej","family":"Ros\u0142anowski","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200018521_ref011","doi-asserted-by":"publisher","DOI":"10.2140\/pjm.1984.115.399"},{"key":"S0022481200018521_ref015","unstructured":"[RS2] Ros\u0142anowski A. and Shelah S. , Simple forcing notions and forcing axioms, preprint."},{"key":"S0022481200018521_ref007","doi-asserted-by":"publisher","DOI":"10.4064\/fm-144-1-23-42"},{"key":"S0022481200018521_ref006","volume-title":"Was G\u00f6del right?","author":"Judah","year":"1989"},{"key":"S0022481200018521_ref001","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(92)90027-W"},{"key":"S0022481200018521_ref010","first-page":"78","volume":"54","author":"Judah","year":"1989","journal-title":"Martin's axioms, measurability and equiconsistency results"},{"key":"S0022481200018521_ref005","volume-title":"Set theory","author":"Jech","year":"1978"},{"key":"S0022481200018521_ref004","doi-asserted-by":"publisher","DOI":"10.1305\/ndjfl\/1093870823"},{"key":"S0022481200018521_ref009","unstructured":"[JS2] Judah H. and Shelah S. , Souslin absoluteness, preprint."},{"key":"S0022481200018521_ref003","first-page":"1323","volume":"58","author":"Goldstern","year":"1993","journal-title":"Strong measure zero sets without Cohen reals"},{"key":"S0022481200018521_ref013","doi-asserted-by":"publisher","DOI":"10.1007\/BF02760523"},{"key":"S0022481200018521_ref014","volume-title":"Archive for Mathematical Logic","author":"Roslanowski"},{"key":"S0022481200018521_ref016","first-page":"153","article-title":"How special are Cohen and random forcings","volume":"88","author":"Shelah","year":"1994","journal-title":"Israel Journal of Mathematics"},{"key":"S0022481200018521_ref002","doi-asserted-by":"publisher","DOI":"10.1016\/0166-8641(90)90112-F"},{"key":"S0022481200018521_ref017","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-21543-2"},{"key":"S0022481200018521_ref008","first-page":"1182","volume":"53","author":"Judah","year":"1988","journal-title":"Souslin forcing"},{"key":"S0022481200018521_ref020","doi-asserted-by":"publisher","DOI":"10.1016\/0001-8708(92)90038-M"},{"key":"S0022481200018521_ref019","doi-asserted-by":"publisher","DOI":"10.4153\/CJM-1991-048-8"},{"key":"S0022481200018521_ref012","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(70)90009-4"},{"key":"S0022481200018521_ref018","volume-title":"Boolean algebras","author":"Sikorski","year":"1964"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200018521","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,13]],"date-time":"2019-05-13T21:52:44Z","timestamp":1557784364000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200018521\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1995,6]]},"references-count":20,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1995,6]]}},"alternative-id":["S0022481200018521"],"URL":"https:\/\/doi.org\/10.2307\/2275837","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1995,6]]}}}