{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,6,19]],"date-time":"2025-06-19T05:51:00Z","timestamp":1750312260840},"reference-count":2,"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":26286,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1942,3,24]]},"abstract":"<jats:p>This note is concerned with the logical formalism with types recently introduced by Church [1] (and called (C) in this note) It was shewn in his paper (Theorem 26<jats:sup>\u03b1<\/jats:sup>) that if <jats:italic>Y<\/jats:italic><jats:sup>\u03b1<\/jats:sup> stands for<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S002248120003704X_Uequ1\" \/><\/jats:disp-formula><\/jats:p><jats:p>(a form of the \u201caxiom of infinity\u201d for the type <jats:italic>\u03b1<\/jats:italic>), <jats:italic>Y<jats:sup>\u03b1<\/jats:sup><\/jats:italic> can be proved formally, from <jats:italic>Y<jats:sup>\u03b9<\/jats:sup><\/jats:italic> and the axioms 1 to 7, for all types <jats:italic>\u03b1<\/jats:italic> of the forms <jats:italic>\u03b9\u2032<\/jats:italic>, <jats:italic>\u03b9\u2033<\/jats:italic>, \u2026. For other types the question was left open, but for the purposes of an intrinsic characterisation of the Church type-stratification given by one of us, it is desirable to have the remaining cases cleared up. A formal proof of <jats:italic>Y<jats:sup>\u03b1<\/jats:sup><\/jats:italic> is now given for all types <jats:italic>\u03b1<\/jats:italic> containing <jats:italic>\u03b9<\/jats:italic>, but the proof uses, in addition to Axioms 1 to 7 and <jats:italic>Y<jats:sup>\u03b9<\/jats:sup><\/jats:italic>, also Axiom 9 (in connection with Def. 4), and Axiom 10 (in Theorem 9).<\/jats:p>","DOI":"10.2307\/2267552","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:44:49Z","timestamp":1146941089000},"page":"28-33","source":"Crossref","is-referenced-by-count":13,"title":["A formal theorem in Church's theory of types"],"prefix":"10.1017","volume":"7","author":[{"given":"M. H. A.","family":"Newman","sequence":"first","affiliation":[]},{"given":"A. M.","family":"Turing","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S002248120003704X_ref002","unstructured":"M. H. A. Newman [1]. Stratified systems of logic. Forthcoming."},{"key":"S002248120003704X_ref001","first-page":"56","article-title":"A formulation of the simple theory of types","volume":"5","author":"A.","year":"1940","journal-title":"this JOURNAL"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S002248120003704X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,6,8]],"date-time":"2019-06-08T11:45:41Z","timestamp":1559994341000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S002248120003704X\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1942,3,24]]},"references-count":2,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1942,3,24]]}},"alternative-id":["S002248120003704X"],"URL":"https:\/\/doi.org\/10.2307\/2267552","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1942,3,24]]}}}