{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,1]],"date-time":"2022-04-01T13:28:12Z","timestamp":1648819692486},"reference-count":9,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":12976,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1978,9]]},"abstract":"<jats:p>Chang, in [1], proves an interpolation theorem (Theorem I, remark b)) for a first-order language. The proof of Chang's theorem uses essentially nonsimple devices, like special and \u03c9<jats:sub>1<\/jats:sub>-saturated models.<\/jats:p><jats:p>In remark e) in [1], Chang asks if there is a simpler proof of his Theorem I.<\/jats:p><jats:p>In [1], Chang proves also another interpolation theorem (Theorem II), which is not an extension of his Theorem I, but extends Craig's interpolation theorem to <jats:italic>L<\/jats:italic><jats:sub>\u03b1<\/jats:sub>+,<jats:sub>\u03c9<\/jats:sub> languages with interpolant in <jats:italic>L<\/jats:italic><jats:sub>\u03b1<\/jats:sub>+,<jats:sub>\u03b1<\/jats:sub> where \u03b1 is a strong limit cardinal of cofinality \u03c9.<\/jats:p><jats:p>In remark k) in [1], Chang asks if there is a generalization of both Theorems I and II in [1], or at least a generalization of both Theorem I in [1] and Lopez-Escobar's interpolation theorem in [7].<\/jats:p><jats:p>Maehara and Takeuti, in [8], show that there is a completely different proof of Chang's interpolation Theorem I as a consequence of their interpolation theorems. The proofs of these theorems of Maehara and Takeuti are proof theoretical in character, involving the notion of cut-free natural deduction, and it uses devices as simple as those needed for the usual Craig's interpolation theorem. Hence this can be considered as a positive answer to Chang's question in remark e) in [1].<\/jats:p>","DOI":"10.2307\/2273530","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:48:21Z","timestamp":1146952101000},"page":"535-549","source":"Crossref","is-referenced-by-count":1,"title":["Interpolation theorems for <i>L<\/i><sub><i>k,k<\/i><\/sub><sup>2+<\/sup>"],"prefix":"10.1017","volume":"43","author":[{"given":"Ruggero","family":"Ferro","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200049422_ref008","first-page":"262","volume":"36","author":"Maehara","year":"1971","journal-title":"Two interpolation theorems for a positive second order predicate calculus"},{"key":"S0022481200049422_ref007","doi-asserted-by":"publisher","DOI":"10.4064\/fm-57-3-253-272"},{"key":"S0022481200049422_ref005","doi-asserted-by":"publisher","DOI":"10.1090\/pspum\/025\/0376341"},{"key":"S0022481200049422_ref004","first-page":"343","volume":"37","author":"Garland","year":"1972","journal-title":"Generalized interpolation theorems"},{"key":"S0022481200049422_ref003","first-page":"123","article-title":"Consistency property and model existence theorem for second order negative languages with conjunctions and quantifications over sets of cardinality smaller than a strong limit cardinal of denumerable cofinality","volume":"55","author":"Ferro","year":"1976","journal-title":"Rendiconti del Seminario Matematico dell' Universita di Padova"},{"key":"S0022481200049422_ref001","first-page":"5","volume-title":"Proceedings of the Rome Conference on Model Theory, Symposia Mathematica","volume":"V","author":"Chang","year":"1970"},{"key":"S0022481200049422_ref006","volume-title":"Model theory for infinitary languages","author":"Keisler","year":"1971"},{"key":"S0022481200049422_ref009","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0079692"},{"key":"S0022481200049422_ref002","first-page":"201","article-title":"Limits to some interpolation theorems","volume":"54","author":"Ferro","year":"1975","journal-title":"Rendiconti del Seminario Matematico dell'Universita di Padova"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200049422","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,27]],"date-time":"2019-05-27T19:33:46Z","timestamp":1558985626000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200049422\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1978,9]]},"references-count":9,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1978,9]]}},"alternative-id":["S0022481200049422"],"URL":"https:\/\/doi.org\/10.2307\/2273530","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1978,9]]}}}