{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,29]],"date-time":"2025-09-29T11:55:03Z","timestamp":1759146903599},"reference-count":19,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":6585,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1996,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>The context for this paper is Feferman's theory of explicit mathematics, <jats:bold>T<jats:sub>0<\/jats:sub><\/jats:bold>. We address a problem that was posed in [6]. Let <jats:bold>MID<\/jats:bold> be the principle stating that any monotone operation on classifications has a least fixed point. The main objective of this paper is to show that <jats:bold>T<jats:sub>0<\/jats:sub><\/jats:bold> + <jats:bold>MID<\/jats:bold>, when based on classical logic, also proves the existence of non-monotone inductive definitions that arise from arbitrary extensional operations on classifications. From the latter we deduce that <jats:bold>MID<\/jats:bold>, when adjoined to classical <jats:bold>T<jats:sub>0<\/jats:sub><\/jats:bold>, leads to a much stronger theory than <jats:bold>T<jats:sub>0<\/jats:sub><\/jats:bold>.<\/jats:p>","DOI":"10.2307\/2275601","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:57:36Z","timestamp":1146941856000},"page":"125-146","source":"Crossref","is-referenced-by-count":7,"title":["Monotone inductive definitions in explicit mathematics"],"prefix":"10.1017","volume":"61","author":[{"given":"Michael","family":"Rathjen","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200017679_ref005","first-page":"159","volume-title":"Logic colloquium '78","author":"Feferman","year":"1979"},{"key":"S0022481200017679_ref016","doi-asserted-by":"publisher","DOI":"10.1007\/BF01621475"},{"key":"S0022481200017679_ref010","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1976-14194-8"},{"key":"S0022481200017679_ref014","volume-title":"Elementary inductions on abstract structures","author":"Moschovakis","year":"1974"},{"key":"S0022481200017679_ref008","unstructured":"Gla\u03b2 T. , Standardstrukturen f\u00fcr Systeme expliziter Mathematik, Thesis, University of M\u00fcnster, 1993."},{"key":"S0022481200017679_ref004","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0062852"},{"key":"S0022481200017679_ref001","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(08)71120-0"},{"key":"S0022481200017679_ref007","first-page":"78","volume-title":"Iterated inductive definitions and subsystems of analysis","author":"Feferman","year":"1981"},{"key":"S0022481200017679_ref018","first-page":"301","volume-title":"Generalized recursion theory","author":"Richter","year":"1974"},{"key":"S0022481200017679_ref002","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-68952-9"},{"key":"S0022481200017679_ref003","first-page":"78","volume-title":"Lecture notes in mathematics","author":"Buchholz","year":"1981"},{"key":"S0022481200017679_ref009","doi-asserted-by":"publisher","DOI":"10.1007\/BF01278464"},{"key":"S0022481200017679_ref011","first-page":"8","volume-title":"Inductive definability","author":"Harrington"},{"key":"S0022481200017679_ref012","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(08)71119-4"},{"key":"S0022481200017679_ref013","volume-title":"Stanford Report on the Foundations of Analysis","author":"Kreisel","year":"1963"},{"key":"S0022481200017679_ref017","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(08)71233-3"},{"key":"S0022481200017679_ref019","first-page":"255","volume-title":"Annals of Pure and Applied Logic","author":"Takahashi","year":"1989"},{"key":"S0022481200017679_ref015","volume-title":"Untersuchungen zu Teilsystemen der Zahlentheorie zweiter Stufe und der Mengenlehre mit einer zwischen  \u2212 CA und \u2212 CA + BI liegenden Beweisst\u00e4rke","author":"Rathjen","year":"1989"},{"key":"S0022481200017679_ref006","first-page":"77","volume-title":"The L. E. J. Brouwer centenary symposium","author":"Feferman","year":"1982"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200017679","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,13]],"date-time":"2019-05-13T15:07:32Z","timestamp":1557760052000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200017679\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1996,3]]},"references-count":19,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1996,3]]}},"alternative-id":["S0022481200017679"],"URL":"https:\/\/doi.org\/10.2307\/2275601","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1996,3]]}}}