{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,15]],"date-time":"2026-06-15T22:36:58Z","timestamp":1781563018689,"version":"3.54.5"},"reference-count":20,"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":10054,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1986,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Let <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200030760_inline1\"\/> be any proper ideal of subsets of the real line <jats:italic>R<\/jats:italic> which contains all finite subsets of <jats:italic>R<\/jats:italic>. We define an ideal <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200030760_inline1\"\/>* \u2223 <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200030760_inline2\"\/> as follows: <jats:italic>X<\/jats:italic> \u2208 <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200030760_inline1\"\/>* \u2223 <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200030760_inline2\"\/> if there exists a Borel set <jats:italic>B<\/jats:italic> \u2282 <jats:italic>R<\/jats:italic> \u00d7 <jats:italic>R<\/jats:italic> such that <jats:italic>X<\/jats:italic> \u2282 <jats:italic>B<\/jats:italic> and for any <jats:italic>x<\/jats:italic> \u2208 <jats:italic>R<\/jats:italic> we have {<jats:italic>y<\/jats:italic> \u2208 <jats:italic>R<\/jats:italic>: \u3008<jats:italic>x, y<\/jats:italic>\u3009 \u2208 <jats:italic>B<\/jats:italic>} \u2208 <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200030760_inline1\"\/>. We show that there exists a family <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200030760_inline3\"\/> \u2282 <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200030760_inline1\"\/>* \u2223 <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200030760_inline2\"\/> of power \u03c9<jats:sub>1<\/jats:sub> such that \u22c3<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200030760_inline3\"\/> \u2209 <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200030760_inline1\"\/>* \u2223 <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200030760_inline2\"\/>.<\/jats:p><jats:p>In the last section we investigate properties of ideals of Lebesgue measure zero sets and meager sets in Cohen extensions of models of set theory.<\/jats:p>","DOI":"10.2307\/2274013","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:18:32Z","timestamp":1146953912000},"page":"560-569","source":"Crossref","is-referenced-by-count":15,"title":["On ideals of subsets of the plane and on Cohen reals"],"prefix":"10.1017","volume":"51","author":[{"given":"Jacek","family":"Cicho\u0144","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Janusz","family":"Pawlikowski","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200030760_ref001","doi-asserted-by":"publisher","DOI":"10.2307\/1999530"},{"key":"S0022481200030760_ref017","volume-title":"Boolean algebras","author":"Sikorski","year":"1964"},{"key":"S0022481200030760_ref014","volume-title":"Descriptive set theory","author":"Moschovakis","year":"1980"},{"key":"S0022481200030760_ref016","doi-asserted-by":"publisher","DOI":"10.4064\/fm-30-1-215-217"},{"key":"S0022481200030760_ref011","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(70)90009-4"},{"key":"S0022481200030760_ref010","volume-title":"Set theory","author":"Kuratowski","year":"1976"},{"key":"S0022481200030760_ref005","unstructured":"Fremlin D. , On Cicho\u0144's diagrams, preprint, 1984."},{"key":"S0022481200030760_ref009","volume-title":"Topologie","volume":"1","author":"Kuratowski","year":"1958"},{"key":"S0022481200030760_ref006","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511896972"},{"key":"S0022481200030760_ref015","author":"Pawlikowski","journal-title":"Why Solovay real produces Cohen real"},{"key":"S0022481200030760_ref012","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1981-0613787-2"},{"key":"S0022481200030760_ref002","doi-asserted-by":"publisher","DOI":"10.1016\/0001-8708(80)90057-2"},{"key":"S0022481200030760_ref020","first-page":"1009","volume":"48","author":"Truss","year":"1983","journal-title":"The noncommutativity of random and generic extensions"},{"key":"S0022481200030760_ref019","first-page":"595","volume-title":"Logic Colloquium 76","author":"Truss","year":"1977"},{"key":"S0022481200030760_ref008","unstructured":"Krawczyk A. , A note on random reals, preprint, 1983."},{"key":"S0022481200030760_ref004","unstructured":"Cicho\u0144 J. , On random reals and the measure algebra (to appear)."},{"key":"S0022481200030760_ref007","volume-title":"Open days for model theory and set theory, Jadwisin 1981","author":"Koepke"},{"key":"S0022481200030760_ref003","volume-title":"Open days for model theory and set theory, Jadwisin 1981","author":"Cicho\u0144"},{"key":"S0022481200030760_ref018","doi-asserted-by":"publisher","DOI":"10.2307\/1970696"},{"key":"S0022481200030760_ref013","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1984-0735576-9"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200030760","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,22]],"date-time":"2019-05-22T04:40:55Z","timestamp":1558500055000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200030760\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1986,9]]},"references-count":20,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1986,9]]}},"alternative-id":["S0022481200030760"],"URL":"https:\/\/doi.org\/10.2307\/2274013","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1986,9]]}}}