{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,8,20]],"date-time":"2023-08-20T23:11:44Z","timestamp":1692573104374},"reference-count":7,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":10693,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1984,12]]},"abstract":"<jats:p>E. Specker has proved that the axiom of choice (AC) is false in NF [6]. Since AC is stratified, one can, according to another famous result of Specker [7], prove directly \u00acAC in type theory (TT) plus some finite set of ambiguity axioms, i.e. sentences of the form <jats:italic>\u03c6<\/jats:italic> \u2194 <jats:italic>\u03c6<\/jats:italic><jats:sup>+<\/jats:sup>, where <jats:italic>\u03c6<\/jats:italic><jats:sup>+<\/jats:sup> results from <jats:italic>\u03c6<\/jats:italic> by adding one to its type indices.<\/jats:p><jats:p>We shall in \u00a72 of this paper give a disproof of AC directly in TT plus some axioms of ambiguity. The argument will be split into two parts. The first one (contained in Proposition 2) concerns cardinal arithmetic and has nothing to do with typical ambiguity. Though carried out in TT, it could have been done in other set theories such as Zermelo's Z or ZF. The second part is an application of this to the cardinals of the universes at different types. This is made possible through the introduction of an appropriate definition of 2<jats:sup><jats:italic>\u03b1<\/jats:italic><\/jats:sup> in \u00a71 enabling one to express shifting sentences as \u201ctyped properties\u201d of the universe, in Boffa's sense. The disproof of AC is then completed in TT plus two extra ambiguity axioms. In \u00a73, we show that this is in a sense the best possible result: that means that every single ambiguity axiom is <jats:italic>consistent<\/jats:italic> with TT plus AC, thus giving a positive solution to a conjecture of Specker [7, p. 119].<\/jats:p>","DOI":"10.2307\/2274262","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:10:48Z","timestamp":1146953448000},"page":"1074-1078","source":"Crossref","is-referenced-by-count":3,"title":["Typical ambiguity and the axiom of choice"],"prefix":"10.1017","volume":"49","author":[{"given":"Marcel","family":"Crabb\u00e9","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200042377_ref007","first-page":"116","volume-title":"Logic, Methodology, and Philosophy of Science: Proceedings of the 1960 International Congress","author":"Specker","year":"1962"},{"key":"S0022481200042377_ref006","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.39.9.972"},{"key":"S0022481200042377_ref005","doi-asserted-by":"publisher","DOI":"10.1090\/pspum\/013.1\/0280359"},{"key":"S0022481200042377_ref004","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19790250703"},{"key":"S0022481200042377_ref001","first-page":"149","volume":"40","author":"Boffa","year":"1975","journal-title":"Sets equipollent to their power set in NF"},{"key":"S0022481200042377_ref003","volume-title":"Quine's new foundations","author":"Forster","year":"1983"},{"key":"S0022481200042377_ref002","first-page":"17","volume-title":"La th\u00e9orie des ensembles de Quine","author":"Crabb\u00e9","year":"1982"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200042377","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,23]],"date-time":"2019-05-23T19:11:12Z","timestamp":1558638672000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200042377\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1984,12]]},"references-count":7,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1984,12]]}},"alternative-id":["S0022481200042377"],"URL":"https:\/\/doi.org\/10.2307\/2274262","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1984,12]]}}}