{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,1]],"date-time":"2022-04-01T06:58:47Z","timestamp":1648796327553},"reference-count":6,"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<jats:sub>1<\/jats:sub>-saturated nonstandard universe a cut is an initial segment of the hyperinlegers 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<\/jats:italic>, <jats:italic>x<\/jats:italic> + <jats:italic>y<\/jats:italic>] \u2286 O. 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=\"S0022481200022799_inline1\" \/>, which is a hyperfinite initial segment of the hyperintegers. <jats:italic>U<\/jats:italic> is called a good cut if there exists a <jats:italic>U<\/jats:italic>-meager subset of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200022799_inline1\" \/> of Loeb measure one. Otherwise <jats:italic>U<\/jats:italic> is bad. In this paper we discuss the questions of Keisler and Leth about the existence of bad cuts and related cuts. We show that assuming <jats:bold>b<\/jats:bold> &gt; \u03c9<jats:sub>1<\/jats:sub>, every hyperfinite time line has a cut with both cofinality and coinitiality uncountable. We construct bad cuts in a nonstandard universe under ZFC. We also give two results about the existence of other kinds of cuts.<\/jats:p>","DOI":"10.2307\/2275286","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:44:45Z","timestamp":1146955485000},"page":"522-527","source":"Crossref","is-referenced-by-count":3,"title":["Cuts 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":"S0022481200022799_ref002","volume-title":"Model theory","author":"Chang","year":"1973"},{"key":"S0022481200022799_ref001","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(88)90048-6"},{"key":"S0022481200022799_ref003","first-page":"1167","volume":"54","author":"Keisler","year":"1989","journal-title":"Descriptive set theory over hyperfinite sets"},{"key":"S0022481200022799_ref004","first-page":"71","volume":"56","author":"Keisler","year":"1991","journal-title":"Meager sets on the hyperfinite time line"},{"key":"S0022481200022799_ref005","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1975-0390154-8"},{"key":"S0022481200022799_ref006","volume-title":"Foundations of infinitesimal stochastic analysis","author":"Stroyan","year":"1986"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200022799","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,16]],"date-time":"2019-05-16T21:45:16Z","timestamp":1558043116000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200022799\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1992,6]]},"references-count":6,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1992,6]]}},"alternative-id":["S0022481200022799"],"URL":"https:\/\/doi.org\/10.2307\/2275286","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1992,6]]}}}