{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,11,14]],"date-time":"2023-11-14T10:55:12Z","timestamp":1699959312731},"reference-count":13,"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":14346,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1974,12]]},"abstract":"<jats:p>In this paper we define by means of a partition property a decreasing sequence <jats:italic>N<\/jats:italic> = \u2039<jats:italic>N<\/jats:italic><jats:sub><jats:italic>\u03b1<\/jats:italic><\/jats:sub>: <jats:italic>\u03b1<\/jats:italic> is an ordinal\u203a of classes of ordinals. This property is a generalization of the nonexistence of special Aronszajn trees: the successor cardinal <jats:italic>\u03ba<\/jats:italic><jats:sup>+<\/jats:sup> is in <jats:italic>N<\/jats:italic><jats:sub>0<\/jats:sub> iff there does not exist a special Aronszajn <jats:italic>\u03ba<\/jats:italic><jats:sup>+<\/jats:sup>-tree.<\/jats:p><jats:p>The interest in the classes <jats:italic>N<\/jats:italic><jats:sub><jats:italic>\u03b1<\/jats:italic><\/jats:sub> stems from their applicability in model theory, in particular to that aspect of model theory dealing with ordered and two-cardinal models. A model <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200077215_inline01\" \/> is <jats:italic>\u03ba-like<\/jats:italic> iff &lt; is a linear ordering of <jats:italic>A<\/jats:italic> of cardinality <jats:italic>\u03ba<\/jats:italic> but such that every proper initial segment has cardinality &lt; <jats:italic>\u03ba<\/jats:italic>. <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200077215_inline02\" \/> is <jats:italic>\u03b1-ordered<\/jats:italic> iff \u227c is a reflexive, linear ordering of some subset of <jats:italic>A<\/jats:italic> with order type <jats:italic>\u03b1<\/jats:italic>. The sequence <jats:italic>N<\/jats:italic> can be characterized by a first-order sentence <jats:italic>\u03c3<\/jats:italic> in the following manner: The sentence <jats:italic>\u03c3<\/jats:italic> has a <jats:italic>\u03ba<\/jats:italic>-like <jats:italic>\u03b1<\/jats:italic>-ordered model iff <jats:italic>\u03ba<\/jats:italic> \u2209 <jats:italic>N<jats:sub>\u03b1<\/jats:sub><\/jats:italic>. This characterization will allow us to translate various independence statements regarding the sequence <jats:italic>N<\/jats:italic> to statements about the independence of transfer properties. We say that the transfer property <jats:italic>\u03ba \u2192 \u03bb<\/jats:italic> holds iff every first-order sentence which has a <jats:italic>\u03ba<\/jats:italic>-like model also has a <jats:italic>\u03bb<\/jats:italic>-like model. <jats:italic>\u03ba<\/jats:italic> \u21f8 <jats:italic>\u03bb<\/jats:italic> is the negation of <jats:italic>\u03ba<\/jats:italic> \u2192 <jats:italic>\u03bb<\/jats:italic>.<\/jats:p>","DOI":"10.2307\/2272857","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:32:58Z","timestamp":1146936778000},"page":"732-740","source":"Crossref","is-referenced-by-count":4,"title":["Generalizing special Aronszajn trees"],"prefix":"10.1017","volume":"39","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":"S0022481200077215_ref006","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(72)90001-0"},{"key":"S0022481200077215_ref011","first-page":"521","volume":"37","author":"Schmerl","year":"1972","journal-title":"An elementary sentence which has ordered models"},{"key":"S0022481200077215_ref009","doi-asserted-by":"publisher","DOI":"10.7146\/math.scand.a-10648"},{"key":"S0022481200077215_ref005","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1956-10036-0"},{"key":"S0022481200077215_ref008","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(72)90017-4"},{"key":"S0022481200077215_ref004","first-page":"93","article-title":"Partition relations for cardinal numbers","volume":"16","author":"Erd\u00f6s","year":"1965","journal-title":"Acta Mathematica"},{"key":"S0022481200077215_ref002","unstructured":"Boos William , Boolean extensions which efface the Mahlo property (mimeographed)."},{"key":"S0022481200077215_ref003","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1965-0193016-3"},{"key":"S0022481200077215_ref010","unstructured":"Schmerl J. H. , On \u03ba-like models for inaccessible \u03ba, Doctoral Dissertation, University of California, Berkeley, 1971."},{"key":"S0022481200077215_ref012","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1974-0337617-8"},{"key":"S0022481200077215_ref007","volume-title":"Memoirs of the American Mathematical Society","author":"L\u00e9vy","year":"1965"},{"key":"S0022481200077215_ref001","first-page":"226","volume":"34","author":"Barwise","year":"1969","journal-title":"Infinitary logic and admissible sets"},{"key":"S0022481200077215_ref013","first-page":"531","volume":"37","author":"Schmerl","year":"1972","journal-title":"On power-like models for hyperinaccessible cardinals"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200077215","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,29]],"date-time":"2019-05-29T16:42:47Z","timestamp":1559148167000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200077215\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1974,12]]},"references-count":13,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1974,12]]}},"alternative-id":["S0022481200077215"],"URL":"https:\/\/doi.org\/10.2307\/2272857","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1974,12]]}}}