{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,3]],"date-time":"2022-04-03T06:17:24Z","timestamp":1648966644520},"reference-count":9,"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":6859,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1995,6]]},"abstract":"<jats:p>The <jats:italic>\u03ba<\/jats:italic>-isomorphism property (IP<jats:italic><jats:sub>\u03ba<\/jats:sub><\/jats:italic>) for nonstandard universes was introduced by Henson in [4]. There has been some recent effort aimed at more fully understanding this property. Jin and Shelah in [7] have shown that for <jats:italic>\u03ba<\/jats:italic> &lt; \u2290<jats:italic><jats:sub>\u03c9<\/jats:sub><\/jats:italic>, IP<jats:italic><jats:sub>\u03ba<\/jats:sub><\/jats:italic> is equivalent to what we will refer to as the <jats:italic>\u03ba<\/jats:italic>-resplendence property. Earlier, in [6], Jin asked if IP<jats:italic><jats:sub>\u03ba<\/jats:sub><\/jats:italic> is equivalent to IP<jats:sub>\u21350<\/jats:sub> plus <jats:italic>\u03ba<\/jats:italic>-saturation. He answered this question positively for <jats:italic>\u03ba<\/jats:italic> = \u2135<jats:sub>1<\/jats:sub>. In this note we extend this answer to all <jats:italic>\u03ba<\/jats:italic>. We also extend the result of Jin and Shelah to all <jats:italic>\u03ba<\/jats:italic>. (Jin also observed this could be done.)<\/jats:p><jats:p>In order to strike a balance between the generalities of model theory and the specifics of nonstandard analysis, we will consider models of Zermelo set theory with the Axiom of Choice; we denote this theory by ZC. The axioms of ZC are just those of ZFC but without the replacement scheme. Thus, among the axioms of ZC are the power set axiom, the infinity axiom, the separation axioms and the axiom of choice.<\/jats:p><jats:p>Let(<jats:italic>V, E<\/jats:italic>) \u22a8 ZC. If <jats:italic>a<\/jats:italic> \u2208 <jats:italic>V<\/jats:italic>, we let *<jats:italic>a<\/jats:italic> = {<jats:italic>x<\/jats:italic> \u2208 <jats:italic>V<\/jats:italic>: (<jats:italic>V,E<\/jats:italic>) \u22a8 <jats:italic>x<\/jats:italic> \u2208 <jats:italic>a<\/jats:italic>}. In particular, <jats:italic>i<\/jats:italic> \u2208 *<jats:italic>\u03c9<\/jats:italic> iff <jats:italic>i<\/jats:italic> \u2208 <jats:italic>V<\/jats:italic> and (<jats:italic>V,E<\/jats:italic>) \u22a8 (<jats:italic>i<\/jats:italic> is a natural number). A subset <jats:italic>A<\/jats:italic> \u2286 <jats:italic>V<\/jats:italic> is <jats:italic>internal<\/jats:italic> if <jats:italic>A<\/jats:italic> = *<jats:italic>a<\/jats:italic> for some <jats:italic>a<\/jats:italic> \u2208 <jats:italic>V<\/jats:italic>.<\/jats:p><jats:p>The <jats:italic>standard model<\/jats:italic> of ZC consists of those sets of rank at most <jats:italic>\u03c9<\/jats:italic> + <jats:italic>\u03c9<\/jats:italic>. In other words, if we let <jats:italic>V<\/jats:italic><jats:sub>0<\/jats:sub> be the set of hereditarily finite sets and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200018612_inline1\" \/> for <jats:italic>n<\/jats:italic> &lt; <jats:italic>\u03c9<\/jats:italic>, then (<jats:italic>V<jats:sub>\u03c9<\/jats:sub><\/jats:italic>, \u2208) is the standard model of ZC, where <jats:italic>V<jats:sub>\u03c9<\/jats:sub><\/jats:italic> = \u22c3<jats:sub><jats:italic>n&lt;\u03c9<\/jats:italic><\/jats:sub>. <jats:italic>V<jats:sub>n<\/jats:sub><\/jats:italic>.<\/jats:p>","DOI":"10.2307\/2275846","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:55:05Z","timestamp":1146956105000},"page":"512-516","source":"Crossref","is-referenced-by-count":4,"title":["The isomorphism property for nonstandard universes"],"prefix":"10.1017","volume":"60","author":[{"given":"James H.","family":"Schmerl","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200018612_ref006","first-page":"1011","volume":"57","author":"Jin","year":"1992","journal-title":"A theorem on the isomorphism property"},{"key":"S0022481200018612_ref009","unstructured":"[9] Schmerl J. H. , A reflection property and its application to nonstandard models, this Journal (to appear)."},{"key":"S0022481200018612_ref004","first-page":"717","volume":"39","author":"Henson","year":"1974","journal-title":"The isomorphism property in nonstandard analysis and its use in the theory of Banach spaces"},{"key":"S0022481200018612_ref007","first-page":"292","volume":"59","author":"Jin","year":"1994","journal-title":"The strength of the isomorphism property"},{"key":"S0022481200018612_ref001","first-page":"531","volume":"41","author":"Barwise","year":"1976","journal-title":"An introduction to recursively saturated and resplendent models"},{"key":"S0022481200018612_ref002","first-page":"470","volume":"49","author":"Buechler","year":"1984","journal-title":"Expansions of models of co-stable theories"},{"key":"S0022481200018612_ref003","volume-title":"Model theory","author":"Chang","year":"1990"},{"key":"S0022481200018612_ref005","first-page":"975","volume":"57","author":"Jin","year":"1992","journal-title":"The isomorphism property versus the special model axiom"},{"key":"S0022481200018612_ref008","first-page":"1016","volume":"56","author":"Keisler","year":"1991","journal-title":"Making the hyperreal line both saturated and complete"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200018612","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,13]],"date-time":"2019-05-13T21:52:38Z","timestamp":1557784358000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200018612\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1995,6]]},"references-count":9,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1995,6]]}},"alternative-id":["S0022481200018612"],"URL":"https:\/\/doi.org\/10.2307\/2275846","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1995,6]]}}}