{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,31]],"date-time":"2022-03-31T06:04:07Z","timestamp":1648706647707},"reference-count":17,"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":8593,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1990,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In this paper we ask the question: to what extent do basic set theoretic properties of Loeb measure depend on the nonstandard universe and on properties of the model of set theory in which it lies? We show that, assuming Martin's axiom and \u03ba-saturation, the smallest cover by Loeb measure zero sets must have cardinality less than \u03ba. In contrast to this we show that the additivity of Loeb measure cannot be greater than <jats:italic>\u03c9<\/jats:italic><jats:sub>1<\/jats:sub>. Define cof(H) as the smallest cardinality of a family of Loeb measure zero sets which cover every other Loeb measure zero set. We show that card(\u230alog<jats:sub>2<\/jats:sub>(<jats:italic>H<\/jats:italic>)\u230b) \u2264 cof (<jats:italic>H)<\/jats:italic> \u2264 card(2<jats:sup><jats:italic>H<\/jats:italic><\/jats:sup>), where card is the external cardinality. We answer a question of Paris and Mills concerning cuts in nonstandard models of number theory. We also present a pair of nonstandard universes <jats:italic>M<\/jats:italic> \u227c <jats:italic>N<\/jats:italic> and hyperfinite integer <jats:italic>H<\/jats:italic> \u2208 <jats:italic>M<\/jats:italic> such that <jats:italic>H<\/jats:italic> is not enlarged by <jats:italic>N<\/jats:italic>, 2<jats:sup><jats:italic>H<\/jats:italic><\/jats:sup> contains new elements, but every new subset of <jats:italic>H<\/jats:italic> has Loeb measure zero. We show that it is consistent that there exists a Sierpi\u0144ski set in the reals but no Loeb-Sierpi\u0144ski set in any nonstandard universe. We also show that it is consistent with the failure of the continuum hypothesis that Loeb-Sierpi\u0144ski sets can exist in some nonstandard universes and even in an ultrapower of a standard universe.<\/jats:p>","DOI":"10.2307\/2274471","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:36:50Z","timestamp":1146940610000},"page":"1022-1036","source":"Crossref","is-referenced-by-count":2,"title":["Set theoretic properties of Loeb measure"],"prefix":"10.1017","volume":"55","author":[{"given":"Arnold W.","family":"Miller","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200025536_ref012","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1975-0390154-8"},{"key":"S0022481200025536_ref003","doi-asserted-by":"publisher","DOI":"10.1112\/blms\/15.6.529"},{"key":"S0022481200025536_ref002","volume-title":"Model thoery","author":"Chang","year":"1973"},{"key":"S0022481200025536_ref004","volume-title":"An introduction to probability theory and its applications","volume":"I","author":"Feller","year":"1968"},{"key":"S0022481200025536_ref008","first-page":"1167","article-title":"Descriptive set theory over hyperfinite sets","volume":"54","author":"Keisler","year":"1989","journal-title":"Journal"},{"key":"S0022481200025536_ref001","doi-asserted-by":"publisher","DOI":"10.2307\/2048302"},{"key":"S0022481200025536_ref005","doi-asserted-by":"publisher","DOI":"10.4153\/CJM-1979-066-0"},{"key":"S0022481200025536_ref014","first-page":"275","volume":"47","author":"Miller","year":"1982","journal-title":"The Baire category theorem and cardinals of countable cofinality"},{"key":"S0022481200025536_ref011","first-page":"253","volume":"48","author":"Kotlarski","year":"1983","journal-title":"On cofinal extensions of models of arithmetic"},{"key":"S0022481200025536_ref006","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1979-0521888-9"},{"key":"S0022481200025536_ref007","first-page":"47","volume":"32","author":"Keisler","year":"1967","journal-title":"Ultraproducts of finite sets"},{"key":"S0022481200025536_ref009","unstructured":"Keisler H. J. and Leth S. , Meager sets on the hyperfinite time line (to appear)."},{"key":"S0022481200025536_ref010","first-page":"129","volume":"52","author":"Kaufmann","year":"1987","journal-title":"Remarks on weak notions of saturation in models of Peano arithmetic"},{"key":"S0022481200025536_ref013","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(77)90006-7"},{"key":"S0022481200025536_ref015","doi-asserted-by":"publisher","DOI":"10.4064\/fm-103-3-205-215"},{"key":"S0022481200025536_ref016","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1975-0357105-9"},{"key":"S0022481200025536_ref017","first-page":"83","volume":"35","author":"Shelah","year":"1970","journal-title":"On the cardinality of ultraproduct of finite sets"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200025536","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,18]],"date-time":"2019-05-18T16:25:44Z","timestamp":1558196744000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200025536\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1990,9]]},"references-count":17,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1990,9]]}},"alternative-id":["S0022481200025536"],"URL":"https:\/\/doi.org\/10.2307\/2274471","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1990,9]]}}}