{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,12,30]],"date-time":"2022-12-30T18:53:35Z","timestamp":1672426415792},"reference-count":9,"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":17908,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1965,3]]},"abstract":"<jats:p>In this paper, by a <jats:italic>function of ordinals<\/jats:italic> we understand a function which is defined for all ordinals and each of whose value is an ordinal. In [7] (also cf. [8] or [9]) we defined <jats:italic>recursive functions and predicates<\/jats:italic> of ordinals, following Kleene's definition on natural numbers. A predicate will be called <jats:italic>arithmetical<\/jats:italic>, if it is obtained from a recursive predicate by prefixing a sequence of alternating quantifiers. A function will be called <jats:italic>arithmetical<\/jats:italic>, if its representing predicate is arithmetical.<\/jats:p><jats:p>The cardinals are identified with those ordinals <jats:italic>a<\/jats:italic> which have larger power than all smaller ordinals than <jats:italic>a<\/jats:italic>. For any given ordinal <jats:italic>a<\/jats:italic>, we denote by <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200057078_inline1\" \/> the cardinal of <jats:italic>a<\/jats:italic> and by 2<jats:sup><jats:italic>a<\/jats:italic><\/jats:sup> the cardinal which is of the same power as the power set of <jats:italic>a<\/jats:italic>. Let \u03c7 be the function such that \u03c7(<jats:italic>a<\/jats:italic>) is the least cardinal which is greater than <jats:italic>a<\/jats:italic>.<\/jats:p><jats:p>Now there are functions of ordinals such that they are easily defined in set theory, but it seems impossible to define them as arithmetical ones; \u03c7 is such a function. If we define \u03c7 in making use of only the language on ordinals, it seems necessary to use the notion of <jats:italic>all<\/jats:italic> the functions from ordinals, e.g., as in [6].<\/jats:p>","DOI":"10.2307\/2270575","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T20:27:20Z","timestamp":1146947240000},"page":"1-7","source":"Crossref","is-referenced-by-count":8,"title":["Transcendence of cardinals"],"prefix":"10.1017","volume":"30","author":[{"given":"Gaisi","family":"Takeuti","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200057078_ref007","doi-asserted-by":"publisher","DOI":"10.2969\/jmsj\/01220119"},{"key":"S0022481200057078_ref008","doi-asserted-by":"publisher","DOI":"10.2969\/jmsj\/01420199"},{"key":"S0022481200057078_ref005","doi-asserted-by":"publisher","DOI":"10.2969\/jmsj\/00620196"},{"key":"S0022481200057078_ref006","doi-asserted-by":"publisher","DOI":"10.2969\/jmsj\/00910093"},{"key":"S0022481200057078_ref009","unstructured":"Takeuti G. , A formalization of the theory of ordinal numbers. To appear in this Journal."},{"key":"S0022481200057078_ref001","volume-title":"The consistency of the axiom of choice and of the generalized continuum hypothesis with the axioms of set theory","author":"G\u00f6del","year":"1940"},{"key":"S0022481200057078_ref002","doi-asserted-by":"publisher","DOI":"10.2140\/pjm.1960.10.223"},{"key":"S0022481200057078_ref003","first-page":"187","article-title":"\u00dcber lineare transfinite Mengen","volume":"63","author":"Mahlo","year":"1911","journal-title":"Berichte \u00fcber die Verhandlungen der K\u00f6niglich S\u00e4chsischen Gesellschaft der Wissenschaften zu Leipzig"},{"key":"S0022481200057078_ref004","first-page":"108","article-title":"Zur Theorie and Anwendung der \u03c10-Zahlen","volume":"64","author":"Mahlo","year":"1912","journal-title":"Berichte \u00fcber die Verhandlungen der K\u00f6niglich S\u00e4chsischen Gesellschaft der Wissenschaften zu Leipzig"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200057078","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,6,3]],"date-time":"2019-06-03T19:50:14Z","timestamp":1559591414000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200057078\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1965,3]]},"references-count":9,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1965,3]]}},"alternative-id":["S0022481200057078"],"URL":"https:\/\/doi.org\/10.2307\/2270575","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1965,3]]}}}