{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,6,5]],"date-time":"2025-06-05T12:05:58Z","timestamp":1749125158312},"reference-count":14,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":6676,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1995,12]]},"abstract":"<jats:p>Some methods of constructing nonstandard models work only for particular theories, such as ZFC, or CA + AC (which is second order number theory with the choice scheme). The examples of this which motivated the results of this paper occur in the main theorems of [5], which state that if <jats:italic>T<\/jats:italic> is any consistent extension of either ZFC<jats:sub>0<\/jats:sub> (which is ZFC but with only countable replacement) or CA + AC and if <jats:italic>\u03ba<\/jats:italic> and <jats:italic>\u03bb<\/jats:italic> are suitably chosen cardinals, then <jats:italic>T<\/jats:italic> has a model which is <jats:italic>\u03ba<\/jats:italic>-saturated and has the <jats:italic>\u03bb<\/jats:italic>-Bolzano-Weierstrass property. (Compare with Theorem 3.5.) Another example is a result from [12] which states that if <jats:italic>T<\/jats:italic> is any consistent extension of CA + AC and cf (<jats:italic>\u03bb<\/jats:italic>) &gt; \u2135<jats:sub>0<\/jats:sub>, then <jats:italic>T<\/jats:italic> has a natural <jats:italic>\u03bb<\/jats:italic>-Archimedean model. (Compare with Theorem 3.1 and the comments following it.) Still another example is a result in [6] in which it is shown that if a model <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200017965_inline1\" \/> of Peano arithmetic is expandable to a model of ZF or of CA, then so is any cofinal extension of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200017965_inline1\" \/>. (Compare with Theorem 3.10.) Related types of constructions can also be found in [10] and [11].<\/jats:p><jats:p>A reflection principle will be proved here, allowing these constructions to be extended to models of many other theories, among which are some exceedingly weak theories and also all of their completions.<\/jats:p>","DOI":"10.2307\/2275878","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:56:55Z","timestamp":1146956215000},"page":"1137-1152","source":"Crossref","is-referenced-by-count":1,"title":["A reflection principle and its applications to nonstandard models"],"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":"S0022481200017965_ref009","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1974-0337617-8"},{"key":"S0022481200017965_ref013","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1986-0846594-0"},{"key":"S0022481200017965_ref004","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1987-0869404-5"},{"key":"S0022481200017965_ref011","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0090952"},{"key":"S0022481200017965_ref010","first-page":"732","volume":"39","author":"Schmerl","year":"1974","journal-title":"Generalizing special Aronszajn trees"},{"key":"S0022481200017965_ref007","doi-asserted-by":"publisher","DOI":"10.2140\/pjm.1973.46.523"},{"key":"S0022481200017965_ref006","first-page":"253","volume":"48","author":"Kotlarski","year":"1983","journal-title":"On cofinal extensions of models of arithmetic"},{"key":"S0022481200017965_ref005","first-page":"1016","volume":"56","author":"Keisler","year":"1991","journal-title":"Making the hyperreal line both saturated and complete"},{"key":"S0022481200017965_ref003","first-page":"888","volume":"47","author":"Devlin","year":"1982","journal-title":"The combinatorial principle \u25ca*"},{"key":"S0022481200017965_ref002","volume-title":"Model theory","author":"Chang","year":"1990"},{"key":"S0022481200017965_ref001","first-page":"251","volume":"45","author":"Beller","year":"1980","journal-title":"A strengthening of Jensen's \u25a1 principles"},{"key":"S0022481200017965_ref014","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(82)90005-5"},{"key":"S0022481200017965_ref008","doi-asserted-by":"publisher","DOI":"10.2140\/pjm.1981.92.195"},{"key":"S0022481200017965_ref012","doi-asserted-by":"publisher","DOI":"10.1007\/BF02761121"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200017965","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,13]],"date-time":"2019-05-13T20:18:34Z","timestamp":1557778714000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200017965\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1995,12]]},"references-count":14,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1995,12]]}},"alternative-id":["S0022481200017965"],"URL":"https:\/\/doi.org\/10.2307\/2275878","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1995,12]]}}}