{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,1]],"date-time":"2026-03-01T14:10:53Z","timestamp":1772374253393,"version":"3.50.1"},"reference-count":10,"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":2018,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2008,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In 2004 Csima, Hirschfeldt, Knight, and Soare [1] showed that a set <jats:italic>A<\/jats:italic> \u2264<jats:sub>T<\/jats:sub> 0\u2032 is nonlow<jats:sub>2<\/jats:sub> if and only if <jats:italic>A<\/jats:italic> is <jats:italic>prime bounding<\/jats:italic>, i.e., for every complete atomic decidable theory <jats:italic>T<\/jats:italic>, there is a prime model <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200004230_inline1\"\/> computable in <jats:italic>A<\/jats:italic>. The authors presented nine seemingly unrelated predicates of a set <jats:italic>A<\/jats:italic>, and showed that they are equivalent for <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200004230_inline2\"\/> sets. Some of these predicates, such as prime bounding, and others involving equivalence structures and abelian <jats:italic>p<\/jats:italic>-groups come from model theory, while others involving meeting dense sets in trees and escaping a given function come from pure computability theory.<\/jats:p><jats:p>As predicates of <jats:italic>A<\/jats:italic>, the original nine properties are equivalent for <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200004230_inline2\"\/> sets; however, they are not equivalent in general. This article examines the (degree-theoretic) relationship between the nine properties. We show that the nine properties fall into three classes, each of which consists of several equivalent properties. We also investigate the relationship between the three classes, by determining whether or not any of the predicates in one class implies a predicate in another class.<\/jats:p>","DOI":"10.2178\/jsl\/1230396753","type":"journal-article","created":{"date-parts":[[2008,12,27]],"date-time":"2008-12-27T11:52:45Z","timestamp":1230378765000},"page":"885-905","source":"Crossref","is-referenced-by-count":5,"title":["Classifying model-theoretic properties"],"prefix":"10.1017","volume":"73","author":[{"given":"Chris J.","family":"Conidis","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200004230_ref006","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1305\/ndjfl\/1039724885","article-title":"Computable models of theories with few models","volume":"38","author":"Khoussainov","year":"1997","journal-title":"Notre Dame Journal of Formal Logic"},{"key":"S0022481200004230_ref009","first-page":"164","volume":"65","author":"Shinoda","year":"2000","journal-title":"Recursive in a generic real"},{"key":"S0022481200004230_ref002","unstructured":"Hirschfeldt D. R. , Shore R. A. , and Slaman T. A. , The atomic model theorem, to appear."},{"key":"S0022481200004230_ref001","first-page":"1117","volume":"69","author":"Csima","year":"2004","journal-title":"Bounding prime models"},{"key":"S0022481200004230_ref003","first-page":"51","volume-title":"Izvestiya Akademii Nauk Kazakhsko\u01d0 SSR. Seriya Fiziko-Matematicheskaya","author":"Khisamiev","year":"1981"},{"key":"S0022481200004230_ref004","doi-asserted-by":"publisher","DOI":"10.1007\/BF00969170"},{"key":"S0022481200004230_ref007","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-21755-9"},{"key":"S0022481200004230_ref008","doi-asserted-by":"crossref","first-page":"307","DOI":"10.1305\/ndjfl\/1022615611","article-title":"A new spectrum of recursive models","volume":"40","author":"Nies","year":"1999","journal-title":"Notre Dame Journal of Formal Logic"},{"key":"S0022481200004230_ref005","first-page":"1177","volume-title":"Handbook of recursive mathematics","volume":"138","author":"Khisamiev","year":"1998"},{"key":"S0022481200004230_ref010","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-02460-7"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200004230","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,30]],"date-time":"2019-04-30T18:32:39Z","timestamp":1556649159000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200004230\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,9]]},"references-count":10,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2008,9]]}},"alternative-id":["S0022481200004230"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1230396753","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,9]]}}}