{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,6,11]],"date-time":"2022-06-11T23:08:04Z","timestamp":1654988884071},"reference-count":1,"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":6585,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1996,3]]},"abstract":"<jats:p>Let <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200017746_inline1\" \/> denote the ideal of Lebesgue measure zero subsets of the real line. Then add(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200017746_inline1\" \/>) denotes the minimal cardinality of a subset of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200017746_inline1\" \/> whose union is not an element of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200017746_inline1\" \/>. In [1] Bartoszynski gave an elegant combinatorial characterization of add(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200017746_inline1\" \/>), namely: add(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200017746_inline1\" \/>) is the least cardinal number <jats:italic>\u03ba<\/jats:italic> for which the following assertion fails:<\/jats:p><jats:p><jats:italic>For every family <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200017746_inline2\" \/> of at most<\/jats:italic><jats:italic>\u03ba functions from \u03c9 to \u03c9 there is a function F from \u03c9 to the finite subsets of \u03c9 such that:<\/jats:italic><\/jats:p><jats:p>1. <jats:italic>For each m, F(m) has at most m<\/jats:italic> + 1 <jats:italic>elements, and<\/jats:italic><\/jats:p><jats:p>2. <jats:italic>for each f in<\/jats:italic><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200017746_inline2\" \/><jats:italic>there are only finitely many m such that f(m) is not an element of F(m)<\/jats:italic>.<\/jats:p><jats:p>The symbol A(<jats:italic>\u03ba<\/jats:italic>) will denote the assertion above about <jats:italic>\u03ba<\/jats:italic>. In the course of his proof, Bartoszynski also shows that the cardinality restriction in 1 is not sharp. Indeed, let (<jats:italic>R<jats:sub>m<\/jats:sub><\/jats:italic>: <jats:italic>m<\/jats:italic> &lt; <jats:italic>\u03c9<\/jats:italic>) be any sequence of integers such that for each <jats:italic>m R<jats:sub>m<\/jats:sub><\/jats:italic>, \u2264 <jats:italic>R<\/jats:italic><jats:sub><jats:italic>m<\/jats:italic>+1<\/jats:sub>, and such that lim<jats:sub><jats:italic>m<\/jats:italic>\u2192\u221e<\/jats:sub><jats:italic>R<jats:sub>m<\/jats:sub><\/jats:italic> = \u221e. Then the truth of the assertion A(<jats:italic>\u03ba<\/jats:italic>) is preserved if in 1 we say instead that<\/jats:p><jats:p>1\u2032. <jats:italic>For each m, F(m) has at most R<jats:sub>m<\/jats:sub> elements<\/jats:italic>.<\/jats:p><jats:p>We shall use this observation later on. We now define three more statements, denoted B(<jats:italic>\u03ba<\/jats:italic>), C(<jats:italic>\u03ba<\/jats:italic>) and D(<jats:italic>\u03ba<\/jats:italic>), about cardinal number <jats:italic>\u03ba<\/jats:italic>.<\/jats:p>","DOI":"10.2307\/2275608","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:57:36Z","timestamp":1146956256000},"page":"246-249","source":"Crossref","is-referenced-by-count":2,"title":["Lebesque measure zero subsets of the real line and an infinite game"],"prefix":"10.1017","volume":"61","author":[{"given":"Marion","family":"Scheepers","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200017746_ref001","doi-asserted-by":"publisher","DOI":"10.2307\/1999530"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200017746","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,13]],"date-time":"2019-05-13T19:07:47Z","timestamp":1557774467000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200017746\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1996,3]]},"references-count":1,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1996,3]]}},"alternative-id":["S0022481200017746"],"URL":"https:\/\/doi.org\/10.2307\/2275608","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1996,3]]}}}