{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,28]],"date-time":"2025-10-28T00:26:20Z","timestamp":1761611180379},"reference-count":4,"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":11699,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1982,3]]},"abstract":"<jats:p>Let \u2018<jats:italic>\u03d5<\/jats:italic>\u2019, \u2018\u03c7\u2019, and \u2018\u03c8\u2019 be variables ranging over functions from the sentence letters <jats:italic>P<\/jats:italic><jats:sub>0<\/jats:sub>, <jats:italic>P<\/jats:italic><jats:sub>1<\/jats:sub>, \u2026 <jats:italic>P<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>, \u2026 of (propositional) modal logic to sentences of P(eano) Arithmetic), and for each sentence <jats:italic>A<\/jats:italic> of modal logic, inductively define <jats:italic>A<\/jats:italic><jats:sup><jats:italic>\u03d5<\/jats:italic><\/jats:sup> by<\/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=\"S0022481200044820_eqnU01\" \/><\/jats:disp-formula><\/jats:p><jats:p>[and similarly for other nonmodal propositional connectives]; and<\/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=\"S0022481200044820_eqnU02\" \/><\/jats:disp-formula><\/jats:p><jats:p>where Bew(<jats:italic>x<\/jats:italic>) is the standard provability predicate for PA and \u2308<jats:italic>F<\/jats:italic>\u2309 is the PA numeral for the G\u00f6del number of the formula <jats:italic>F<\/jats:italic> of PA. Then for any <jats:italic>\u03d5<\/jats:italic>, (\u2212\u25a1\u22a5)<jats:sup><jats:italic>\u03d5<\/jats:italic><\/jats:sup> = \u2212Bew(\u2308\u22a5\u2309), which is the consistency assertion for PA; a sentence <jats:italic>S<\/jats:italic> is undecidable in PA iff both <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200044820_inline01\" \/> and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200044820_inline02\" \/>, where <jats:italic>\u03d5<\/jats:italic>(<jats:italic>p<\/jats:italic><jats:sub>0<\/jats:sub>) = <jats:italic>S<\/jats:italic>. If <jats:italic>\u03c8<\/jats:italic>(<jats:italic>p<\/jats:italic><jats:sub>0<\/jats:sub>) is the undecidable sentence constructed by G\u00f6del, then \u22ac<jats:sub>PA<\/jats:sub> (\u2212\u25a1\u22a5\u2192 \u2212\u25a1<jats:italic>p<\/jats:italic><jats:sub>0<\/jats:sub> &amp; \u2212 \u25a1 \u2212 <jats:italic>p<\/jats:italic><jats:sub>0<\/jats:sub>)<jats:sup><jats:italic>\u03c8<\/jats:italic><\/jats:sup> and \u22a2<jats:sub>PA<\/jats:sub>(<jats:italic>P<\/jats:italic><jats:sub>0<\/jats:sub> \u2194 \u2212\u25a1\u22a5)<jats:sup><jats:italic>\u03c8<\/jats:italic><\/jats:sup>. However, if <jats:italic>\u03c8<\/jats:italic>(<jats:italic>p<\/jats:italic><jats:sub>0<\/jats:sub>) is the undecidable sentence constructed by Rosser, then the situation is the other way around: \u22ac<jats:sub>PA<\/jats:sub>(<jats:italic>P<\/jats:italic><jats:sub>0<\/jats:sub> \u2194 \u2212\u25a1\u22a5)<jats:sup><jats:italic>\u03c8<\/jats:italic><\/jats:sup> and \u22a2<jats:sub>PA<\/jats:sub> (\u2212\u25a1\u22a5\u2192 \u2212\u25a1\u2212<jats:italic>p<\/jats:italic><jats:sub>0<\/jats:sub> &amp; \u2212\u25a1\u2212<jats:italic>p<\/jats:italic><jats:sub>0<\/jats:sub>)<jats:italic>\u03c8<\/jats:italic>. We call a sentence <jats:italic>S<\/jats:italic> of PA <jats:italic>extremely undecidable<\/jats:italic> if for all modal sentences <jats:italic>A<\/jats:italic> containing no sentence letter other than <jats:italic>p<\/jats:italic><jats:sub>0<\/jats:sub>, if for some <jats:italic>\u03c8<\/jats:italic>, \u22ac<jats:sub>PA<\/jats:sub><jats:italic>A<\/jats:italic><jats:sup><jats:italic>\u03c8<\/jats:italic><\/jats:sup>, then \u22ac<jats:sub>PA<\/jats:sub><jats:italic>A<\/jats:italic><jats:sup><jats:italic>\u03d5<\/jats:italic><\/jats:sup>, where <jats:italic>\u03d5<\/jats:italic>(<jats:italic>p<\/jats:italic><jats:sub>0<\/jats:sub>) = <jats:italic>S<\/jats:italic>. (So, roughly speaking, a sentence is extremely undecidable if it can be proved to have only those modal-logically characterizable properties that every sentence can be proved to have.) Thus extremely undecidable sentences are undecidable, but neither the Godel nor the Rosser sentence is extremely undecidable. It will follow at once from the main theorem of this paper that there are infinitely many inequivalent extremely undecidable sentences.<\/jats:p>","DOI":"10.2307\/2273393","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:58:32Z","timestamp":1146952712000},"page":"191-196","source":"Crossref","is-referenced-by-count":18,"title":["Extremely undecidable sentences"],"prefix":"10.1017","volume":"47","author":[{"given":"George","family":"Boolos","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200044820_ref004","doi-asserted-by":"publisher","DOI":"10.1007\/BF02757006"},{"key":"S0022481200044820_ref003","volume-title":"Metamathematical investigation of intuitionistic arithmetic and analysis","author":"Smorynski","year":"1973"},{"key":"S0022481200044820_ref002","first-page":"647","article-title":"\u2018Flexible\u2019 predicates of formal number theory","volume":"13","author":"Kripke","year":"1962","journal-title":"Proceedings of the American Mathematical Society"},{"key":"S0022481200044820_ref001","volume-title":"The unprovability of consistency","author":"Boolos","year":"1979"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200044820","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,24]],"date-time":"2019-05-24T21:42:29Z","timestamp":1558734149000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200044820\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1982,3]]},"references-count":4,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1982,3]]}},"alternative-id":["S0022481200044820"],"URL":"https:\/\/doi.org\/10.2307\/2273393","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1982,3]]}}}