{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,3]],"date-time":"2022-04-03T21:35:49Z","timestamp":1649021749430},"reference-count":0,"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":23676,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1949,5,16]]},"abstract":"<jats:p>A demonstrably consistent theory of real numbers has been outlined by the writer in <jats:italic>An extension of basic logic<\/jats:italic><jats:sup>1<\/jats:sup> (hereafter referred to as EBL). This theory deals with non-negative real numbers, but it could be easily modified to deal with negative real numbers also. It was shown that the theory was adequate for proving a form of the fundamental theorem on least upper bounds and greatest lower bounds. More precisely, the following results were obtained in the terminology of EBL: If \u0421 is a class of <jats:italic>U<\/jats:italic>-reals and is completely represented in \u039a\u2032 and if some <jats:italic>U<\/jats:italic>-real is an upper bound of \u0421, then there is a <jats:italic>U<\/jats:italic>-real which is a least upper bound of \u0421. If <jats:italic>D<\/jats:italic> is a class of (<jats:italic>U<\/jats:italic>-reals and is completely represented in \u039a\u2032, then there is a <jats:italic>U<\/jats:italic>-real which is a greatest lower bound of <jats:italic>D<\/jats:italic>.<\/jats:p>","DOI":"10.2307\/2268972","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T15:04:01Z","timestamp":1146927841000},"page":"9-15","source":"Crossref","is-referenced-by-count":5,"title":["The Heine-Borel theorem in extended basic logic"],"prefix":"10.1017","volume":"14","author":[{"given":"Frederic B.","family":"Fitch","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200073114","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,6,8]],"date-time":"2019-06-08T06:32:54Z","timestamp":1559975574000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200073114\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1949,5,16]]},"references-count":0,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1949,5,16]]}},"alternative-id":["S0022481200073114"],"URL":"https:\/\/doi.org\/10.2307\/2268972","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1949,5,16]]}}}