{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,5]],"date-time":"2025-10-05T04:33:20Z","timestamp":1759638800055},"reference-count":13,"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":10876,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1984,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Under the assumption that all \u201crules\u201d are recursive (ECT) the statement Cont(<jats:italic>N<jats:sup>N<\/jats:sup>, N<\/jats:italic>) that all functions from <jats:italic>N<jats:sup>N<\/jats:sup><\/jats:italic> to <jats:italic>N<\/jats:italic> are continuous becomes equivalent to a statement KLS in the language of arithmetic about \u201ceffective operations\u201d. Our main result is that KLS is underivable in intuitionistic Zermelo-Fraenkel set theory + ECT. Similar results apply for functions from <jats:italic>R<\/jats:italic> to <jats:italic>R<\/jats:italic> and from 2<jats:sup><jats:italic>N<\/jats:italic><\/jats:sup> to <jats:italic>N<\/jats:italic>. Such results were known for weaker theories, e.g. HA and HAS. We extend not only the theorem but the method, fp-realizability, to intuitionistic ZF.<\/jats:p>","DOI":"10.2307\/2274195","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:08:28Z","timestamp":1146953308000},"page":"630-643","source":"Crossref","is-referenced-by-count":7,"title":["Church's thesis, continuity, and set theory"],"prefix":"10.1017","volume":"49","author":[{"given":"M.","family":"Beeson","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"A.","family":"\u0160\u010dedrov","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200033685_ref013","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(82)90024-9"},{"key":"S0022481200033685_ref011","first-page":"347","volume":"40","author":"Myhill","year":"1975","journal-title":"Constructive set theory"},{"key":"S0022481200033685_ref009","first-page":"290","volume-title":"Constructivity in mathematics (Proceedings of the colloquium held at Amsterdam, 1957)","author":"Kreisel","year":"1959"},{"key":"S0022481200033685_ref008","volume-title":"Lectures on constructive mathematical analysis","author":"Ku\u0161ner","year":"1973"},{"key":"S0022481200033685_ref007","volume-title":"Intensionalmathematics","author":"Goodman"},{"key":"S0022481200033685_ref006","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1975-0373893-X"},{"key":"S0022481200033685_ref004","first-page":"1","article-title":"Continuity in intuitionistic set theories","author":"Beeson","year":"1979","journal-title":"Logic Colloquium '78"},{"key":"S0022481200033685_ref012","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0066739"},{"key":"S0022481200033685_ref001","first-page":"321","volume":"40","author":"Beeson","year":"1975","journal-title":"The nonderivability in intuitionistic formai systems of theorems on the continuity of effective operations"},{"key":"S0022481200033685_ref003","doi-asserted-by":"publisher","DOI":"10.2140\/pjm.1977.68.29"},{"key":"S0022481200033685_ref005","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(82)90003-1"},{"key":"S0022481200033685_ref010","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0066775"},{"key":"S0022481200033685_ref002","first-page":"18","volume":"41","author":"Beeson","year":"1976","journal-title":"The unprovability in intuitionistic formal systems of the continuity of effective operations on the reals"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200033685","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,23]],"date-time":"2019-05-23T20:48:47Z","timestamp":1558644527000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200033685\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1984,6]]},"references-count":13,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1984,6]]}},"alternative-id":["S0022481200033685"],"URL":"https:\/\/doi.org\/10.2307\/2274195","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1984,6]]}}}