{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,31]],"date-time":"2025-12-31T15:53:47Z","timestamp":1767196427660,"version":"build-2238731810"},"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":16082,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1970,3]]},"abstract":"<jats:p>\n                    J. N. Crossley [1] raised the question of whether the implication 2 +\n                    <jats:italic>A = A \u21d2<\/jats:italic>\n                    1 +\n                    <jats:italic>A<\/jats:italic>\n                    =\n                    <jats:italic>A<\/jats:italic>\n                    is true for constructive order types (C.O.T.'s). Using an earlier definition of constructive order type, A. G. Hamilton [2] presented a counterexample. Hamilton left open the general question, however, since he pointed out that Crossley considers only orderings which can be embedded in a standard dense r.e. ordering by a partial recursive function, and that his counterexample fails to meet this requirement. We resolve the question by finding a C.O.T.\n                    <jats:italic>A<\/jats:italic>\n                    which meets Crossley's requirement and such that 2 +\n                    <jats:italic>A<\/jats:italic>\n                    =\n                    <jats:italic>A<\/jats:italic>\n                    but 1 +\n                    <jats:italic>A<\/jats:italic>\n                    \u2260\n                    <jats:italic>A<\/jats:italic>\n                    . At the suggestion of A. B. Manaster and A. G. Hamilton we easily extend this construction to show that for any\n                    <jats:italic>n<\/jats:italic>\n                    \u2267 2, there is a C.O.T.\n                    <jats:italic>A<\/jats:italic>\n                    such that\n                    <jats:italic>n<\/jats:italic>\n                    +\n                    <jats:italic>A<\/jats:italic>\n                    =\n                    <jats:italic>A<\/jats:italic>\n                    but\n                    <jats:italic>m<\/jats:italic>\n                    +\n                    <jats:italic>A<\/jats:italic>\n                    \u2260\n                    <jats:italic>A<\/jats:italic>\n                    for 0 &lt;\n                    <jats:italic>m<\/jats:italic>\n                    &lt;\n                    <jats:italic>n<\/jats:italic>\n                    . Hence, Theorem 3 of [2] and all of its corollaries hold with the new definition of C.O.T. The construction is not difficult and requires no priority argument. The techniques are similar to those developed in [3], but no outside results are needed here.\n                  <\/jats:p>","DOI":"10.2307\/2271163","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T16:58:55Z","timestamp":1146934735000},"page":"119-121","source":"Crossref","is-referenced-by-count":0,"title":["A problem in the theory of constructive order types"],"prefix":"10.1017","volume":"35","author":[{"given":"Robin O.","family":"Gandy","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Robert I.","family":"Soare","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"container-title":["The Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200092306","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,22]],"date-time":"2023-03-22T06:38:01Z","timestamp":1679467081000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200092306\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1970,3]]},"references-count":0,"aliases":["10.1017\/s0022481200092306"],"journal-issue":{"issue":"1","published-print":{"date-parts":[[1970,3]]}},"alternative-id":["S0022481200092306"],"URL":"https:\/\/doi.org\/10.2307\/2271163","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1970,3]]}}}