{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,30]],"date-time":"2022-03-30T10:31:47Z","timestamp":1648636307839},"reference-count":8,"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":7954,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1992,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In an \u03c9-saturated nonstandard universe a cut is an initial segment of the hyperintegers which is closed under addition. Keisler and Leth in [KL] introduced, for each given cut <jats:italic>U<\/jats:italic>, a corresponding <jats:italic>U<\/jats:italic>-topology on the hyperintegers by letting <jats:italic>O<\/jats:italic> be <jats:italic>U<\/jats:italic>-open if for any <jats:italic>x<\/jats:italic> \u03f5 <jats:italic>O<\/jats:italic> there is a <jats:italic>y<\/jats:italic> greater than all the elements in <jats:italic>U<\/jats:italic> such that the interval [<jats:italic>x<\/jats:italic> \u2212 <jats:italic>y,x<\/jats:italic> + <jats:italic>y<\/jats:italic>] \u2286 <jats:italic>O<\/jats:italic>. Let <jats:italic>U<\/jats:italic> be a cut in a hyperfinite time line <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200022805_inline1\" \/>, which is a hyperfinite initial segment of the hyperintegers. A subset B of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200022805_inline1\" \/> is called a <jats:italic>U<\/jats:italic>-Lusin set in <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200022805_inline1\" \/> if <jats:italic>B<\/jats:italic> is uncountable and for any Loeb-Borel <jats:italic>U<\/jats:italic>-meager subset <jats:italic>X<\/jats:italic> of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200022805_inline1\" \/>, <jats:italic>B<\/jats:italic> \u22c2 <jats:italic>X<\/jats:italic> is countable. Here a Loeb-Borel set is an element of the \u03c3-algebra generated by all internal subsets of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200022805_inline1\" \/>: In this paper we answer some questions of Keisler and Leth about the existence of <jats:italic>U<\/jats:italic>-Lusin sets by proving the following facts. (1) If <jats:italic>U<\/jats:italic> = <jats:italic>x<\/jats:italic>\/<jats:bold>N<\/jats:bold> = {<jats:italic>y<\/jats:italic> \u03f5 <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200022805_inline1\" \/>: \u2200<jats:italic>n<\/jats:italic> \u03f5 \u2115(<jats:italic>y<\/jats:italic> &lt; <jats:italic>x<\/jats:italic>\/<jats:italic>n<\/jats:italic>)} for some <jats:italic>x<\/jats:italic> \u03f5 <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200022805_inline1\" \/>, then there exists a <jats:italic>U<\/jats:italic>-Lusin set of power \u03ba if and only if there exists a Lusin set of the reals of power \u03ba. (2) If <jats:italic>U<\/jats:italic> \u2260 <jats:italic>x<\/jats:italic>\/<jats:bold>N<\/jats:bold> but the coinitiality of <jats:italic>U<\/jats:italic> is \u03c9, then there are no <jats:italic>U<\/jats:italic>-Lusin sets if CH fails. (3) Under ZFC there exists a nonstandard universe in which <jats:italic>U<\/jats:italic>-Lusin sets exist for every cut <jats:italic>U<\/jats:italic> with uncountable cofinality and coinitiality. (4) In any \u03c9<jats:sub>2<\/jats:sub>-saturated nonstandard universe there are no <jats:italic>U<\/jats:italic>-Lusin sets for all cuts <jats:italic>U<\/jats:italic> except <jats:italic>U<\/jats:italic> = <jats:italic>x<\/jats:italic>\/<jats:bold>N<\/jats:bold>.<\/jats:p>","DOI":"10.2307\/2275287","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:44:45Z","timestamp":1146955485000},"page":"528-533","source":"Crossref","is-referenced-by-count":0,"title":["<i>U<\/i>-Lusin sets in hyperfinite time lines"],"prefix":"10.1017","volume":"57","author":[{"given":"Renling","family":"Jin","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200022805_ref005","doi-asserted-by":"publisher","DOI":"10.1016\/B978-0-444-86580-9.50008-2"},{"key":"S0022481200022805_ref007","volume-title":"Foundations of infinitesimal stochastic analysis","author":"Stroyan","year":"1986"},{"key":"S0022481200022805_ref008","first-page":"906","volume":"56","author":"\u017divaljevi\u0107","year":"1991","journal-title":"U-meaaer sets when the cofinality and the coinitiality of U are uncountable"},{"key":"S0022481200022805_ref002","first-page":"1167","volume":"54","author":"Keisler","year":"1989","journal-title":"Descriptive set theory over hyperfinite sets"},{"key":"S0022481200022805_ref003","first-page":"71","volume":"56","author":"Keisler","year":"1991","journal-title":"Meager sets on the hyperfinite time line"},{"key":"S0022481200022805_ref001","volume-title":"Model theory","author":"Chang","year":"1973"},{"key":"S0022481200022805_ref004","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1975-0390154-8"},{"key":"S0022481200022805_ref006","first-page":"1022","volume":"55","author":"Miller","year":"1990","journal-title":"Set theoretic properties of Loeb measure"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200022805","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,16]],"date-time":"2019-05-16T21:45:03Z","timestamp":1558043103000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200022805\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1992,6]]},"references-count":8,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1992,6]]}},"alternative-id":["S0022481200022805"],"URL":"https:\/\/doi.org\/10.2307\/2275287","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1992,6]]}}}