{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,10,18]],"date-time":"2023-10-18T15:35:44Z","timestamp":1697643344024},"reference-count":9,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":6493,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1996,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>A structure is <jats:italic>locally finite<\/jats:italic> if every finitely generated substructure is finite; <jats:italic>local<\/jats:italic> sentences are universal sentences all models of which are locally finite. The <jats:italic>stretching theorem<\/jats:italic> for local sentences expresses a remarkable reflection phenomenon between the finite and the infinite models of local sentences. This result in part requires strong axioms to be proved; it was studied by the second named author, in a paper of this Journal, volume 53. Here we correct and extend this paper; in particular we show that the stretching theorem implies the <jats:italic>existence<\/jats:italic> of inaccessible cardinals, and has precisely the <jats:italic>consistency<\/jats:italic> strength of Mahlo cardinals of finite order. And we present a sequel due to the first named author:<\/jats:p><jats:p>(i) decidability of the spectrum Sp(<jats:italic>\u03c6<\/jats:italic>) of a local sentence <jats:italic>\u03c6<\/jats:italic>, below <jats:italic>\u03c9<\/jats:italic><jats:sup><jats:italic>\u03c9<\/jats:italic><\/jats:sup>; where Sp(<jats:italic>\u03c6<\/jats:italic>) is the set of ordinals <jats:italic>\u03b1<\/jats:italic> such that <jats:italic>\u03c6<\/jats:italic> has a model of order type <jats:italic>\u03b1<\/jats:italic><\/jats:p><jats:p>(ii) proof that beth<jats:sub><jats:italic>\u03c9<\/jats:italic><\/jats:sub> = sup{Sp(<jats:italic>\u03c6<\/jats:italic>): <jats:italic>\u03c6<\/jats:italic> local sentence with a bounded spectrum}<\/jats:p><jats:p>(iii) existence of a local sentence <jats:italic>\u03c6<\/jats:italic> such that Sp(<jats:italic>\u03c6<\/jats:italic>) contains all infinite ordinals except the inaccessible cardinals.<\/jats:p>","DOI":"10.2307\/2275676","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:58:29Z","timestamp":1146956309000},"page":"563-585","source":"Crossref","is-referenced-by-count":8,"title":["Stretchings"],"prefix":"10.1017","volume":"61","author":[{"given":"O.","family":"Finkel","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"J. P.","family":"Ressayre","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200017436_ref003","volume-title":"TCS","author":"Finkel","year":"1995"},{"key":"S0022481200017436_ref005","first-page":"991","article-title":"Langages de B\u00fcchi et \u03c9-langages locaux","volume":"1","author":"Finkel","year":"1989","journal-title":"Comptes Rendus de l'Acad\u00e9mie des Sciences, Paris t. 309"},{"key":"S0022481200017436_ref006","unstructured":"Finkel O. , Th\u00e9orie des mod\u00e8les des formules locales et \u00e9tude des langages formels qu'elles d\u00e9finissent, Th\u00e8se de doctorat , Universit\u00e9 Paris 7, Mai 1993."},{"key":"S0022481200017436_ref004","unstructured":"Finkel O. , Propri\u00e9t\u00e9s de cloture des \u03c9-langages localement finis, preprint."},{"key":"S0022481200017436_ref001","volume-title":"Set theory: an introduction to large cardinals","author":"Drake","year":"1974"},{"key":"S0022481200017436_ref009","first-page":"1009","volume":"53","author":"Ressayre","year":"1988","journal-title":"Formal languages defined by the underlying structure of their words"},{"key":"S0022481200017436_ref008","doi-asserted-by":"publisher","DOI":"10.1090\/conm\/065\/891257"},{"key":"S0022481200017436_ref002","doi-asserted-by":"publisher","DOI":"10.4064\/fm-43-1-50-68"},{"key":"S0022481200017436_ref007","volume-title":"Model theory and arithmetic","author":"McAloon","year":"1981"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200017436","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,12]],"date-time":"2019-05-12T21:28:53Z","timestamp":1557696533000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200017436\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1996,6]]},"references-count":9,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1996,6]]}},"alternative-id":["S0022481200017436"],"URL":"https:\/\/doi.org\/10.2307\/2275676","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1996,6]]}}}