{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,31]],"date-time":"2022-03-31T11:36:54Z","timestamp":1648726614763},"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>In this note we investigate an extension of Peano arithmetic which arises from adjoining generalized quantifiers to first-order logic. Markwald [2] first studied the definability properties of <jats:italic>L<\/jats:italic><jats:sup>1<\/jats:sup>, the language of first-order arithmetic, <jats:italic>L<\/jats:italic>, with the additional quantifer <jats:italic>Ux<\/jats:italic> which denotes \u201cthere are infinitely many <jats:italic>x<\/jats:italic> such that\u2026. Note that <jats:italic>Ux<\/jats:italic> is the same thing as the Keisler quantifier <jats:italic>Qx<\/jats:italic> in the \u2135<jats:sub>0<\/jats:sub> interpretation.<\/jats:p><jats:p>We consider <jats:italic>L<\/jats:italic><jats:sup>2<\/jats:sup>, which is <jats:italic>L<\/jats:italic> together with the \u2135<jats:sub>0<\/jats:sub> interpretation of the Magidor-Malitz quantifier <jats:italic>Q<\/jats:italic><jats:sup>2<\/jats:sup><jats:italic>xy<\/jats:italic> which denotes \u201cthere is an infinite set <jats:italic>X<\/jats:italic> such that for distinct <jats:italic>x, y<\/jats:italic> \u2208 <jats:italic>X<\/jats:italic> \u2026\u201d. In [1] Magidor and Malitz presented an axiom system for languages which arise from adding <jats:italic>Q<\/jats:italic><jats:sup>2<\/jats:sup> to a first-order language. They proved that the axioms are valid in every regular interpretation, and, assuming \u25ca<jats:sub><jats:italic>\u03c9<\/jats:italic>1<\/jats:sub>, that the axioms are complete in the \u2135<jats:sub>1<\/jats:sub> interpretation.<\/jats:p><jats:p>If we let <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200044819_inline02\" \/> denote Peano arithmetic in <jats:italic>L<\/jats:italic><jats:sup>2<\/jats:sup> with induction for <jats:italic>L<\/jats:italic><jats:sup>2<\/jats:sup> formulas and the Magidor-Malitz axioms as logical axioms, we show that in <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200044819_inline02\" \/> we can give a truth definition for first-order Peano arithmetic, <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200044819_inline01\" \/>. Consequently we can prove in <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200044819_inline02\" \/> that <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200044819_inline01\" \/> is <jats:italic>\u03a0<jats:sub>n<\/jats:sub><\/jats:italic> sound for every <jats:italic>n<\/jats:italic>, thus in <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200044819_inline02\" \/> we can prove the Paris-Harrington combinatorial principle and the higher-order analogues due to Schlipf.<\/jats:p>","DOI":"10.2307\/2273392","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:58:32Z","timestamp":1146938312000},"page":"187-190","source":"Crossref","is-referenced-by-count":2,"title":["On generalized quantifiers in arithmetic"],"prefix":"10.1017","volume":"47","author":[{"given":"Carl","family":"Morgenstern","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200044819_ref002","doi-asserted-by":"publisher","DOI":"10.1007\/BF01361115"},{"key":"S0022481200044819_ref004","volume-title":"Proof theory","author":"Takeuti","year":"1975"},{"key":"S0022481200044819_ref003","volume-title":"Mathematical logic","author":"Schoenfield","year":"1967"},{"key":"S0022481200044819_ref001","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(77)90019-5"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200044819","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,24]],"date-time":"2019-05-24T17:42:34Z","timestamp":1558719754000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200044819\/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":["S0022481200044819"],"URL":"https:\/\/doi.org\/10.2307\/2273392","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1982,3]]}}}