{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,3]],"date-time":"2022-04-03T04:09:19Z","timestamp":1648958959230},"reference-count":1,"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":12795,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1979,3]]},"abstract":"<jats:p>In order to prove that the Scott's interpolation theorem fails in <jats:italic>L<\/jats:italic><jats:sub>\u03c91\u03c9<\/jats:sub>, H. Africk proved the following lemma in [1]. (See [1] for notations.)<\/jats:p><jats:p>Africk's Lemma. <jats:italic>Suppose that <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048684_inline10\" \/> and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048684_inline20\" \/>. Then every <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048684_inline30\" \/>-sentence in L<jats:sub>\u03c91\u03c9<\/jats:sub> is equivalent to a sentence of the form <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048684_inline50\" \/> and a sentence of the form <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048684_inline40\" \/>, where A<jats:sub>i,j<\/jats:sub>, B<jats:sub>i,j<\/jats:sub>, are F<jats:sub>j<\/jats:sub>-sentences and there are only countable distinct A<jats:sub>i,j<\/jats:sub> or B<jats:sub>i,j<\/jats:sub> together<\/jats:italic>.<\/jats:p><jats:p>But this lemma is false as is shown in the following: Suppose that <jats:italic>Z<\/jats:italic> = {0,1}, <jats:italic>F<jats:sub>j<\/jats:sub><\/jats:italic> = {<jats:italic>P<jats:sub>k,j<\/jats:sub><\/jats:italic>}<jats:sub><jats:italic>k<\/jats:italic>\u2282\u03c9<\/jats:sub> and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048684_inline60\" \/>. Let <jats:italic>A<\/jats:italic> be the sentence \u22c0<jats:sub><jats:italic>k<\/jats:italic>\u2282\u03c9<\/jats:sub>(\u2200<jats:italic>x<\/jats:italic><jats:italic>P<jats:sub>k,0<\/jats:sub><\/jats:italic>(<jats:italic>x<\/jats:italic>) \u2228 \u2200<jats:italic>x<\/jats:italic><jats:italic>P<\/jats:italic><jats:sub><jats:italic>k,1<\/jats:italic><\/jats:sub>(<jats:italic>x<\/jats:italic>)). If Africk's Lemma is true, then there are countably many <jats:italic>F<jats:sub>j<\/jats:sub><\/jats:italic>-sentences <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048684_inline70\" \/>, such that <jats:italic>A<\/jats:italic> is equivalent to <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048684_inline80\" \/> Since <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048684_inline90\" \/> is countable, there are only countably many distinct pairs (<jats:italic>A<jats:sub>i,0<\/jats:sub><\/jats:italic>, <jats:italic>A<jats:sub>i,1<\/jats:sub><\/jats:italic>). So, we get a countable set {(<jats:italic>B<jats:sub>i,0<\/jats:sub><\/jats:italic>, <jats:italic>B<jats:sub>i,1<\/jats:sub><\/jats:italic>)}<jats:sub><jats:italic>i<\/jats:italic>\u2282\u03c9<\/jats:sub>, such that <jats:italic>A<\/jats:italic> is equivalent to \u22c1<jats:sub><jats:italic>i<\/jats:italic>k<\/jats:sub>\u2282\u03c9(<jats:italic>B<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic>,0<\/jats:sub> \u2227 <jats:italic>B<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic>,1<\/jats:sub>). Since <jats:italic>B<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic>,0<\/jats:sub> \u2227 <jats:italic>B<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic>,1<\/jats:sub> \u2192 \u2200<jats:italic>x<\/jats:italic><jats:italic>P<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic>,0<\/jats:sub>(<jats:italic>x<\/jats:italic>) \u2228 \u2200<jats:italic>x<\/jats:italic><jats:italic>P<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic>,1<\/jats:sub> is l-valid(i.e. valid in all first-order structures of cardinality 1) for every <jats:italic>k<\/jats:italic> \u03f5 \u03c9, we have that either <jats:italic>B<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic>,0<\/jats:sub> \u2192 \u2200<jats:italic>x<\/jats:italic><jats:italic>P<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic>,0<\/jats:sub>(<jats:italic>x<\/jats:italic>) is 1-valid or <jats:italic>B<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic>,0<\/jats:sub> \u2192 \u2200<jats:italic>x<\/jats:italic><jats:italic>P<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic>,1<\/jats:sub>(<jats:italic>x<\/jats:italic>) is 1-valid. Let <jats:italic>I<\/jats:italic> be the set of all the <jats:italic>k<\/jats:italic> \u03f5 \u03c9 such that <jats:italic>B<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic>,0<\/jats:sub> \u2192 \u2200<jats:italic>x<\/jats:italic><jats:italic>P<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic>,0<\/jats:sub>(<jats:italic>x<\/jats:italic>) is 1-valid and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048684_inline160\" \/> the first-order structure defined by <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048684_inline100\" \/>, <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048684_inline110\" \/><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048684_inline120\" \/>, <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048684_inline130\" \/> if <jats:italic>k<\/jats:italic> \u03f5 <jats:italic>I<\/jats:italic> and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048684_inline140\" \/>, <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048684_inline150\" \/> if <jats:italic>k<\/jats:italic> \u2209 <jats:italic>I<\/jats:italic>.<\/jats:p>","DOI":"10.2307\/2273700","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:49:34Z","timestamp":1146937774000},"page":"32-32","source":"Crossref","is-referenced-by-count":0,"title":["A remark on Africk's paper on Scott's interpolation theorem for <i>L<\/i><sub>\u03c91\u03c9<\/sub>"],"prefix":"10.1017","volume":"44","author":[{"given":"Nobuyoshi","family":"Motohashi","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200048684_ref001","first-page":"124","volume":"39","author":"Africk","year":"1974","journal-title":"Scott's interpolation theorem fails in L\u03c91\u03c9"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200048684","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,26]],"date-time":"2019-05-26T17:57:03Z","timestamp":1558893423000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200048684\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1979,3]]},"references-count":1,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1979,3]]}},"alternative-id":["S0022481200048684"],"URL":"https:\/\/doi.org\/10.2307\/2273700","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1979,3]]}}}