{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,10,19]],"date-time":"2023-10-19T08:13:11Z","timestamp":1697703191521},"reference-count":7,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2008,12,15]],"date-time":"2008-12-15T00:00:00Z","timestamp":1229299200000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematical Logic Qtrly"],"published-print":{"date-parts":[[2009,1]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>I prove that there is a recursive function <jats:italic>T<\/jats:italic> that does the following: Let <jats:italic>X<\/jats:italic> be transitive and rudimentarily closed, and let <jats:italic>X<\/jats:italic> \u2032 be the closure of <jats:italic>X<\/jats:italic> \u222a {<jats:italic>X<\/jats:italic> } under rudimentary functions. Given a \u03a3<jats:sub>0<\/jats:sub>\u2010formula <jats:italic>\u03c6<\/jats:italic> (<jats:italic>x<\/jats:italic>) and a code <jats:italic>c<\/jats:italic> for a rudimentary function <jats:italic>f<\/jats:italic>, <jats:italic>T<\/jats:italic> (<jats:italic>\u03c6<\/jats:italic>, <jats:italic>c<\/jats:italic>, <jats:styled-content>$ \\vec x $<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/tex2gif-ueqn-1.gif\" xlink:title=\"equation image\" \/><\/jats:styled-content>) is a \u03a3<jats:sub><jats:italic>\u03c9<\/jats:italic> <\/jats:sub>\u2010formula such that for any <jats:styled-content>$ \\vec a $<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/tex2gif-ueqn-2.gif\" xlink:title=\"equation image\" \/><\/jats:styled-content> \u2208 <jats:italic>X<\/jats:italic>, <jats:italic>X<\/jats:italic> \u2032 \u22a7 <jats:italic>\u03c6<\/jats:italic> [<jats:italic>f<\/jats:italic> (<jats:styled-content>$ \\vec a $<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/tex2gif-ueqn-3.gif\" xlink:title=\"equation image\" \/><\/jats:styled-content>)] iff <jats:italic>X<\/jats:italic> \u22a7 <jats:italic>T<\/jats:italic> (<jats:italic>\u03c6<\/jats:italic>, <jats:italic>c<\/jats:italic>, <jats:styled-content>$ \\vec x $<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/tex2gif-ueqn-4.gif\" xlink:title=\"equation image\" \/><\/jats:styled-content>)[<jats:styled-content>$ \\vec a $<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/tex2gif-ueqn-5.gif\" xlink:title=\"equation image\" \/><\/jats:styled-content>]. I make this precise and show relativized versions of this. As an application, I prove that under certain conditions, if <jats:italic>Y<\/jats:italic> is the \u03a3<jats:sub><jats:italic>\u03c9<\/jats:italic> <\/jats:sub> extender ultrapower of <jats:italic>X<\/jats:italic> with respect to some extender <jats:italic>F<\/jats:italic> that also is an extender on <jats:italic>X<\/jats:italic> \u2032, then the closure of <jats:italic>Y<\/jats:italic> \u222a {<jats:italic>Y<\/jats:italic> } under rudimentary functions is the \u03a3<jats:sub>0<\/jats:sub> extender ultrapower of <jats:italic>X\u2032<\/jats:italic> with respect to <jats:italic>F<\/jats:italic>, and the ultrapower embeddings agree on <jats:italic>X<\/jats:italic>. (\u00a9 2009 WILEY\u2010VCH Verlag GmbH &amp; Co. KGaA, Weinheim)<\/jats:p>","DOI":"10.1002\/malq.200710077","type":"journal-article","created":{"date-parts":[[2008,12,15]],"date-time":"2008-12-15T18:01:24Z","timestamp":1229364084000},"page":"4-20","source":"Crossref","is-referenced-by-count":2,"title":["Successor levels of the Jensen hierarchy"],"prefix":"10.1002","volume":"55","author":[{"given":"Gunter","family":"Fuchs","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2008,12,15]]},"reference":[{"key":"e_1_2_1_2_2","doi-asserted-by":"crossref","unstructured":"K. J.Devlin Constructibility (Springer 1984).","DOI":"10.1007\/978-3-662-21723-8"},{"key":"e_1_2_1_3_2","unstructured":"G.Fuchs \u03bb\u2010structures ands\u2010structures: Translating the models. In preparation."},{"key":"e_1_2_1_4_2","unstructured":"G.Fuchs \u03bb\u2010Strukturen unds\u2010Strukturen. Ph. D. dissertation Humboldt\u2010University Berlin Germany Department of Mathematics May 2003. Available athttp:\/\/www.edoc.hu\u2010berlin.de."},{"key":"e_1_2_1_5_2","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(72)90001-0"},{"key":"e_1_2_1_6_2","unstructured":"R.Jensen A new fine structure for higher core models. Handwritten notes 1997."},{"key":"e_1_2_1_7_2","doi-asserted-by":"crossref","unstructured":"W. J.Mitchell andJ. R.Steel Fine Structure and Iteration Trees. Lecture Notes in Logic 3 (Springer 1994).","DOI":"10.1007\/978-3-662-21903-4"},{"key":"e_1_2_1_8_2","doi-asserted-by":"crossref","unstructured":"M.Zeman Inner Models and Large Cardinals (Springer 2002).","DOI":"10.1515\/9783110857818"}],"container-title":["Mathematical Logic Quarterly"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fmalq.200710077","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fmalq.200710077","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/malq.200710077","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,18]],"date-time":"2023-10-18T07:25:56Z","timestamp":1697613956000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/malq.200710077"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,12,15]]},"references-count":7,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2009,1]]}},"alternative-id":["10.1002\/malq.200710077"],"URL":"https:\/\/doi.org\/10.1002\/malq.200710077","archive":["Portico"],"relation":{},"ISSN":["0942-5616","1521-3870"],"issn-type":[{"value":"0942-5616","type":"print"},{"value":"1521-3870","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,12,15]]}}}