{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,8,31]],"date-time":"2023-08-31T04:55:25Z","timestamp":1693457725703},"reference-count":3,"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":14164,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1975,6]]},"abstract":"<jats:p>In this note we define a class of properties for which the following holds: If we can prove in NF that the property holds for the universe <jats:italic>V<\/jats:italic>, then we can prove in NF that it holds for every set equipollent to its power set.<\/jats:p><jats:p>Definition. For any stratified formula <jats:italic>A<\/jats:italic> and any variable \u03c5 which does not occur in <jats:italic>A<\/jats:italic>, let <jats:italic>A<\/jats:italic><jats:sub>\u03c5<\/jats:sub> be the formula obtained by replacing in <jats:italic>A<\/jats:italic> each quantifier (<jats:italic>Qx<\/jats:italic>) by the bounded quantifier (<jats:italic>Qx<\/jats:italic> \u2208 <jats:italic>SC<jats:sup>i<\/jats:sup><\/jats:italic>(<jats:italic>\u03c5<\/jats:italic>)), where <jats:italic>i<\/jats:italic> is the type of <jats:italic>x<\/jats:italic> in <jats:italic>A.<\/jats:italic> We will say that a property <jats:italic>P<\/jats:italic>(<jats:italic>\u03c5<\/jats:italic>) is <jats:italic>typed<\/jats:italic> when there is a stratified sentence <jats:italic>S<\/jats:italic> such that <jats:italic>P<\/jats:italic>(<jats:italic>\u03c5<\/jats:italic>) \u2194 <jats:italic>S<\/jats:italic><jats:sub><jats:italic>\u03c5<\/jats:italic><\/jats:sub> holds in NF.<\/jats:p><jats:p>Examples of typed properties are: \u201c<jats:italic>\u03c5<\/jats:italic> is Dedekind-infinite\u201d, \u201c<jats:italic>\u03c5<\/jats:italic> is not well-orderable\u201d. Specker [3] proved that these typed properties hold for the universe <jats:italic>V<\/jats:italic>, and C. Ward Henson [1] extended this result to any set equipollent to its power set. We will show that such an extension holds for any typed property.<\/jats:p><jats:p>Theorem. <jats:italic>For any typed property P<\/jats:italic>(<jats:italic>\u03c5<\/jats:italic>):<\/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=\"S002248120005369X_eqnU1\" \/><\/jats:disp-formula><\/jats:p><jats:p>Proof. Fix a bijective map <jats:italic>h<\/jats:italic>: <jats:italic>\u03c5<\/jats:italic> \u2192 <jats:italic>SC<\/jats:italic>(<jats:italic>\u03c5<\/jats:italic>) and define for <jats:italic>i<\/jats:italic> = 0, 1, 2, \u2026, <jats:italic>n<\/jats:italic>, \u2026 a bijective map <jats:italic>h<jats:sub>i<\/jats:sub><\/jats:italic>: <jats:italic>\u03c5<\/jats:italic> \u2192 <jats:italic>SC<jats:sup>i<\/jats:sup><\/jats:italic>(<jats:italic>\u03c5<\/jats:italic>) as follows:<\/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=\"S002248120005369X_eqnU2\" \/><\/jats:disp-formula><\/jats:p><jats:p>For every formula <jats:italic>A<\/jats:italic>, let <jats:italic>A<\/jats:italic><jats:sub>(<jats:italic>h<\/jats:italic>)<\/jats:sub> be obtained by replacing in <jats:italic>A<\/jats:italic> each atomic part (<jats:italic>x<\/jats:italic> \u2208 <jats:italic>y<\/jats:italic>) by (<jats:italic>x<\/jats:italic> \u2208 <jats:italic>h<\/jats:italic>(<jats:italic>y<\/jats:italic>)) and each quantifier (<jats:italic>Qx<\/jats:italic>) by (<jats:italic>Qx<\/jats:italic> \u2208 <jats:italic>\u03c5<\/jats:italic>).<\/jats:p>","DOI":"10.2307\/2271894","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:35:13Z","timestamp":1146951313000},"page":"149-150","source":"Crossref","is-referenced-by-count":2,"title":["Sets equipollent to their power set in NF"],"prefix":"10.1017","volume":"40","author":[{"given":"Maurice","family":"Boffa","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S002248120005369X_ref003","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.39.9.972"},{"key":"S002248120005369X_ref001","first-page":"59","volume":"38","author":"Ward Henson","year":"1973","journal-title":"Type-raising operations on cardinal and ordinal numbers in Quine's \u201cNew Foundations\u201d"},{"key":"S002248120005369X_ref002","doi-asserted-by":"publisher","DOI":"10.1007\/BF00568059"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S002248120005369X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,29]],"date-time":"2019-05-29T19:40:42Z","timestamp":1559158842000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S002248120005369X\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1975,6]]},"references-count":3,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1975,6]]}},"alternative-id":["S002248120005369X"],"URL":"https:\/\/doi.org\/10.2307\/2271894","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1975,6]]}}}