{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,2,20]],"date-time":"2024-02-20T11:10:01Z","timestamp":1708427401798},"reference-count":11,"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":4119,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2002,12]]},"abstract":"<jats:p>Some highly saturated models of Peano Arithmetic are constructed in this paper, which consists of two independent sections. In \u00a7 1 we answer a question raised in [10] by constructing some highly saturated, rather classless models of PA. A question raised in [7], [3], ]4] is answered in \u00a72, where highly saturated, nonstandard universes having no bad cuts are constructed.<\/jats:p><jats:p>Highly saturated, rather classless models of Peano Arithmetic were constructed in [10]. The main result proved there is the following theorem. If \u03bb is a regular cardinal and<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200009063_inline1\" \/>is a<jats:italic>\u03bb<\/jats:italic>-saturated model of PA such that \u2223<jats:italic>M<\/jats:italic>\u2223 &gt;<jats:italic>\u03bb<\/jats:italic>, then<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200009063_inline1\" \/>has an elementary extension<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200009063_inline2\" \/>of the same cardinality which is also<jats:italic>\u03bb<\/jats:italic>-saturated and which, in addition, is rather classless. The construction in [10] produced a model<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200009063_inline2\" \/>for which cf(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200009063_inline2\" \/>) =<jats:italic>\u03bb<\/jats:italic><jats:sup>+<\/jats:sup>. We asked in Question 5.1 of [10] what other cofinalities could such a model<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200009063_inline2\" \/>have. This question is answered here in Theorem 1.1 of \u00a71 by showing that any cofinality not immediately excluded is possible. Its proof does not depend on the theorem from [10]; in fact, the proof presented here gives a proof of that theorem which is much simpler and shorter than the one in [10].<\/jats:p><jats:p>Recursively saturated, rather classless \u03ba-like models of PA were constructed in [9]. In the case of singular \u03ba such models were constructed whenever cf(<jats:italic>\u03ba<\/jats:italic>) &gt; \u2135<jats:sub>0<\/jats:sub>; no additional set-theoretic hypothesis was needed.<\/jats:p>","DOI":"10.2178\/jsl\/1190150284","type":"journal-article","created":{"date-parts":[[2007,12,13]],"date-time":"2007-12-13T19:16:54Z","timestamp":1197573414000},"page":"1265-1273","source":"Crossref","is-referenced-by-count":2,"title":["Some highly saturated models of Peano arithmetic"],"prefix":"10.1017","volume":"67","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":"S0022481200009063_ref006","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198532132.001.0001","volume-title":"Models of Peano arithmetic","author":"Kaye","year":"1991"},{"key":"S0022481200009063_ref004","first-page":"1241","volume":"62","author":"Jin","year":"1997","journal-title":"Type two cuts, had cuts and very bad cuts"},{"key":"S0022481200009063_ref002","first-page":"663","volume":"41","author":"Gregory","year":"1976","journal-title":"Higher Souslin trees and the generalized continuum hypothesis"},{"key":"S0022481200009063_ref011","doi-asserted-by":"publisher","DOI":"10.1007\/BF02758648"},{"key":"S0022481200009063_ref003","first-page":"522","volume":"57","author":"Jin","year":"1992","journal-title":"Cuts in the hyperfinite time lines"},{"key":"S0022481200009063_ref005","first-page":"129","volume":"52","author":"Kaufmann","year":"1987","journal-title":"Remarks on weak notions of saturation in models of Peano arithmetic"},{"key":"S0022481200009063_ref008","first-page":"625","volume":"47","author":"Pabion","year":"1982","journal-title":"Saturated models of Peano arithmetic"},{"key":"S0022481200009063_ref009","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0090952"},{"key":"S0022481200009063_ref007","first-page":"71","volume":"56","author":"Keisler","year":"1991","journal-title":"Meager sets on the hyperfinite time line"},{"key":"S0022481200009063_ref010","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-22110-5_10"},{"key":"S0022481200009063_ref001","volume-title":"Model theory","author":"Chang","year":"1990"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200009063","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,2,20]],"date-time":"2024-02-20T10:42:32Z","timestamp":1708425752000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200009063\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2002,12]]},"references-count":11,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2002,12]]}},"alternative-id":["S0022481200009063"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1190150284","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2002,12]]}}}