{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,29]],"date-time":"2022-03-29T19:26:46Z","timestamp":1648582006386},"reference-count":18,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":13068,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1978,6]]},"abstract":"<jats:p>Let Th be a formal theory extending number theory. Call an ordinal \u03be provable in Th if there is a primitive recursive ordering which can be proved in Th to be a wellordering and whose order type is &gt; \u03be. One may define the ordinal \u2223 Th \u2223 of Th to be the least ordinal which is not provable in Th. By [3] and [12] we get <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049525_inline1\" \/>, where ID<jats:sub><jats:italic>N<\/jats:italic><\/jats:sub> is the formal theory for <jats:italic>N<\/jats:italic>-times iterated inductive definitions. We will generalize this result not only to the case of transfinite iterations but also to a more general notion of \u2018the ordinal of a theory\u2019.<\/jats:p><jats:p>For an X-positive arithmetic formula <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049525_inline2\" \/>[X,<jats:italic>x<\/jats:italic>] there is a natural norm \u2223<jats:italic>x<\/jats:italic>\u2223: = \u03bc\u03be (<jats:italic>x<\/jats:italic> \u2208 <jats:italic>I<\/jats:italic><jats:sub arrange=\"stack\"><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049525_inline2\" \/><\/jats:sub><jats:sup arrange=\"stack\">\u03be<\/jats:sup>), where <jats:italic>I<\/jats:italic><jats:sub arrange=\"stack\"><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049525_inline2\" \/><\/jats:sub><jats:sup arrange=\"stack\"><jats:italic>\u03be<\/jats:italic><\/jats:sup> is defined as {<jats:italic>x<\/jats:italic>: <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049525_inline2\" \/>[\u222a<jats:sub><jats:italic>\u03bc<\/jats:italic>&lt;<jats:italic>\u03be<\/jats:italic><\/jats:sub><jats:italic>I<\/jats:italic><jats:sub arrange=\"stack\"><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049525_inline2\" \/><\/jats:sub><jats:sup arrange=\"stack\"><jats:italic>\u03bc<\/jats:italic><\/jats:sup>, <jats:italic>x<\/jats:italic>]}  by recursion on <jats:italic>\u03be<\/jats:italic> (cf. [7], [17]). By P<jats:sup><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049525_inline2\" \/><\/jats:sup> we denote the least fixed point of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049525_inline2\" \/>[<jats:italic>X<\/jats:italic>,<jats:italic>x<\/jats:italic>]. Then P<jats:sup><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049525_inline2\" \/><\/jats:sup> = \u222a<jats:sub><jats:italic>\u03be<\/jats:italic><\/jats:sub><jats:sub arrange=\"stack\"><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049525_inline2\" \/><\/jats:sub><jats:sup arrange=\"stack\"><jats:italic>\u03be<\/jats:italic><\/jats:sup> holds. If Th allows the formation of <jats:italic>P<\/jats:italic><jats:sup><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049525_inline2\" \/><\/jats:sup>, we get the canonical definitions \u2225Th(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049525_inline2\" \/>)\u2225 = sup{\u2223<jats:italic>x<\/jats:italic>\u2223<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049525_inline2\" \/> + 1: Th \u22a2 <jats:italic>x<\/jats:italic> \u2208 <jats:italic>P<\/jats:italic><jats:sup><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049525_inline2\" \/><\/jats:sup>} and \u2225Th\u2225 = sup{\u2225Th(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049525_inline2\" \/>)\u2225: P<jats:sup><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049525_inline2\" \/><\/jats:sup> is definable in Th} (cf. [17]). If \u227a is any primitive recursive ordering define <jats:italic>Q<\/jats:italic>\u227a[<jats:italic>X,x]<\/jats:italic> as the formula \u2200<jats:italic>y<\/jats:italic>(<jats:italic>y<\/jats:italic> \u227a <jats:italic>x<\/jats:italic> \u2192 <jats:italic>y<\/jats:italic> \u2208 <jats:italic>X<\/jats:italic>) and \u2223<jats:italic>x<\/jats:italic>\u2223\u227a:= \u2223<jats:italic>x<\/jats:italic>\u2223<jats:italic>O<\/jats:italic>\u227a. Then \u2223<jats:italic>x<\/jats:italic>\u2223\u227a turns out to be the order type of the \u227a -predecessors of <jats:italic>x<\/jats:italic> and P<jats:sup><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049525_inline2\" \/><\/jats:sup> is the accessible part of \u227a. So Th \u22a2 <jats:italic>x<\/jats:italic> \u2208 <jats:italic>P<\/jats:italic><jats:sup><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049525_inline2\" \/><\/jats:sup> implies the provability of \u2223<jats:italic>x<\/jats:italic>\u2223\u227a in Th.<\/jats:p>","DOI":"10.2307\/2272816","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:47:35Z","timestamp":1146952055000},"page":"161-182","source":"Crossref","is-referenced-by-count":10,"title":["Ordinals connected with formal theories for transfinitely iterated inductive definitions"],"prefix":"10.1017","volume":"43","author":[{"given":"W.","family":"Pohlers","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200049525_ref008","first-page":"190","volume":"18","author":"Myhill","year":"1953","journal-title":"A stumblingblock in constructive mathematics"},{"key":"S0022481200049525_ref013","volume-title":"Beweistheorie","author":"Sch\u00fctte","year":"1960"},{"key":"S0022481200049525_ref005","doi-asserted-by":"publisher","DOI":"10.1007\/BF01565428"},{"key":"S0022481200049525_ref007","volume-title":"Elementary induction on abstract structures","author":"Moschovakis","year":"1974"},{"key":"S0022481200049525_ref016","doi-asserted-by":"publisher","DOI":"10.2307\/1970691"},{"key":"S0022481200049525_ref009","doi-asserted-by":"publisher","DOI":"10.1007\/BF01967653"},{"key":"S0022481200049525_ref004","first-page":"303","volume-title":"Intuitionism and proof theory","author":"Feferman","year":"1970"},{"key":"S0022481200049525_ref001","volume-title":"Proof Theory Symposium","author":"Buchholz","year":"1974"},{"key":"S0022481200049525_ref003","first-page":"118","volume":"43","author":"Buchholz","year":"1978","journal-title":"Provable wellorderings of formal theories for transfinitely iterated inductive definitions"},{"key":"S0022481200049525_ref017","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0066745"},{"key":"S0022481200049525_ref011","unstructured":"Pohlers W. , Eine Grenze f\u00fcr die Herleitbarkeit der transfiniten Induktion in einem schwachen \u03a01 1-Fragment der Analysis, Dissertation, University of M\u00fcnchen, 1973."},{"key":"S0022481200049525_ref012","first-page":"271","volume-title":"Proof Theory Symposium","author":"Pohlers","year":"1974"},{"key":"S0022481200049525_ref018","first-page":"113","volume":"40","author":"Friedman","year":"1975","journal-title":"One hundred and two problems in mathematical logic"},{"key":"S0022481200049525_ref002","doi-asserted-by":"publisher","DOI":"10.1007\/BF02007261"},{"key":"S0022481200049525_ref010","volume-title":"Communications of the Mathematical Institute","author":"Pfeiffer","year":"1973"},{"key":"S0022481200049525_ref014","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-66473-1"},{"key":"S0022481200049525_ref006","doi-asserted-by":"publisher","DOI":"10.2969\/jmsj\/02030456"},{"key":"S0022481200049525_ref015","volume-title":"Mathematical logic","author":"Shoenfield","year":"1967"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200049525","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,27]],"date-time":"2019-05-27T19:57:24Z","timestamp":1558987044000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200049525\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1978,6]]},"references-count":18,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1978,6]]}},"alternative-id":["S0022481200049525"],"URL":"https:\/\/doi.org\/10.2307\/2272816","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1978,6]]}}}