{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,10,24]],"date-time":"2023-10-24T20:59:30Z","timestamp":1698181170231},"reference-count":5,"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":14986,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1973,3]]},"abstract":"<jats:p>In this paper we develop certain methods of proof in Quine's set theory NF which have no counterparts elsewhere. These ideas were first used by Specker [5] in his disproof of the Axiom of Choice in NF. They depend on the properties of two related operations, <jats:italic>T<\/jats:italic>(<jats:italic>n<\/jats:italic>) on cardinal numbers and <jats:italic>U<\/jats:italic>(<jats:italic>\u03b1<\/jats:italic>) on ordinal numbers, which are defined by the equations<\/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=\"S002248120007729X_Uequ1\" \/><\/jats:disp-formula><\/jats:p><jats:p>for each set <jats:italic>x<\/jats:italic> and well ordering <jats:italic>R<\/jats:italic>. (Here and below we use Rosser's notation [3].) The definitions insure that the formulas <jats:italic>T<\/jats:italic>(<jats:italic>x<\/jats:italic>) = <jats:italic>y<\/jats:italic> and <jats:italic>U<\/jats:italic>(<jats:italic>x<\/jats:italic>) = <jats:italic>y<\/jats:italic> are stratified when <jats:italic>y<\/jats:italic> is assigned a type one higher than <jats:italic>x<\/jats:italic>. The importance of <jats:italic>T<\/jats:italic> and <jats:italic>U<\/jats:italic> stems from the following facts: (i) each of <jats:italic>T<\/jats:italic> and <jats:italic>U<\/jats:italic> is a 1-1, order preserving operation from its domain onto a proper initial section of its domain; (ii) Tand <jats:italic>U<\/jats:italic> commute with most of the standard operations on cardinal and ordinal numbers.<\/jats:p><jats:p>These basic facts are discussed in \u00a71. In \u00a72 we prove in NF that the exponential function 2<jats:sup><jats:italic>n<\/jats:italic><\/jats:sup> is not 1-1. Indeed, there exist cardinal numbers <jats:italic>m<\/jats:italic> and <jats:italic>n<\/jats:italic> which satisfy<\/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=\"S002248120007729X_Uequ2\" \/><\/jats:disp-formula><\/jats:p><jats:p>In \u00a73 we prove the following technical result, which has many important applications. Suppose <jats:italic>f<\/jats:italic> is an increasing function from an initial segment <jats:italic>S<\/jats:italic> of the set NO of ordinal numbers into NO and that <jats:italic>f<\/jats:italic> commutes with <jats:italic>U<\/jats:italic>.<\/jats:p>","DOI":"10.2307\/2271726","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:21:18Z","timestamp":1146936078000},"page":"59-68","source":"Crossref","is-referenced-by-count":7,"title":["Type-raising operations on cardinal and ordinal numbers in Quine's \u201cNew foundations\u201d"],"prefix":"10.1017","volume":"38","author":[{"given":"C. Ward","family":"Henson","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S002248120007729X_ref002","first-page":"280","volume":"21","author":"Orey","year":"1956","journal-title":"On the relative consistency of set theory"},{"key":"S002248120007729X_ref001","first-page":"95","volume":"20","author":"Orey","year":"1955","journal-title":"Formal development of ordinal number theory"},{"key":"S002248120007729X_ref004","first-page":"289","volume-title":"Proceedings of the International Congress of Mathematicians","author":"Rosser","year":"1954"},{"key":"S002248120007729X_ref005","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.39.9.972"},{"key":"S002248120007729X_ref003","volume-title":"Logic for mathematicians","author":"Rosser","year":"1953"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S002248120007729X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,30]],"date-time":"2019-05-30T16:08:15Z","timestamp":1559232495000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S002248120007729X\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1973,3]]},"references-count":5,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1973,3]]}},"alternative-id":["S002248120007729X"],"URL":"https:\/\/doi.org\/10.2307\/2271726","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1973,3]]}}}