{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,8]],"date-time":"2026-04-08T09:06:35Z","timestamp":1775639195899,"version":"3.50.1"},"reference-count":12,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":8137,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1991,12]]},"abstract":"<jats:p>We consider the following properties of uncountable-dimensional quadratic spaces (<jats:italic>E<\/jats:italic>, \u03a6):<\/jats:p><jats:p>(*) For all subspaces <jats:italic>U<\/jats:italic> \u2286 <jats:italic>E<\/jats:italic> of infinite dimension: dim <jats:italic>U<\/jats:italic><jats:sup>\u02d4<\/jats:sup> &lt; dim <jats:italic>E<\/jats:italic>.<\/jats:p><jats:p>(**) For all subspaces <jats:italic>U<\/jats:italic> \u2286 <jats:italic>E<\/jats:italic> of infinite dimension: dim <jats:italic>U<\/jats:italic><jats:sup>\u02d4<\/jats:sup> &lt; \u2135<jats:sub>0<\/jats:sub>.<\/jats:p><jats:p>Spaces of countable dimension are the orthogonal sum of straight lines and planes, so they cannot have (*), but (**) is trivially satisfied.<\/jats:p><jats:p>These properties have been considered first in [G\/O] in the process of investigating the orthogonal group of quadratic spaces. It has been shown there (in ZFC) that over arbitrary uncountable fields (**)-spaces of uncountable dimension exist.<\/jats:p><jats:p>In [B\/G], (**)-spaces of dimension \u2135<jats:sub>1<\/jats:sub> (so (*) = (**)) have been constructed over arbitrary finite or countable fields. But this could be done only under the assumption that the continuum hypothesis (CH) holds in the underlying set theory.<\/jats:p>","DOI":"10.2307\/2275468","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:42:28Z","timestamp":1146940948000},"page":"1195-1211","source":"Crossref","is-referenced-by-count":8,"title":["Independence and consistency proofs in quadratic form theory"],"prefix":"10.1017","volume":"56","author":[{"given":"James E.","family":"Baumgartner","sequence":"first","affiliation":[]},{"given":"Otmar","family":"Spinas","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200023549_ref009","unstructured":"Spinas O. , Konsistenz- und Unabh\u00e4ngigkeitsresultate in der Theorie der quadratischen Formen, Ph.D. thesis, University of Z\u00fcrich, Z\u00fcrich, 1989."},{"key":"S0022481200023549_ref011","first-page":"161","article-title":"An undecidability result in lattice theory","volume":"11","author":"Spinas","year":"1990","journal-title":"Abstracts of Papers Presented to the American Mathematical Society"},{"key":"S0022481200023549_ref007","volume-title":"Set theory: an introduction to independence proofs","author":"Kunen","year":"1980"},{"key":"S0022481200023549_ref006","doi-asserted-by":"publisher","DOI":"10.1007\/BF02566137"},{"key":"S0022481200023549_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/BF02567381"},{"key":"S0022481200023549_ref002","doi-asserted-by":"publisher","DOI":"10.1016\/B978-0-444-86580-9.50006-9"},{"key":"S0022481200023549_ref003","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511896972"},{"key":"S0022481200023549_ref005","unstructured":"Gross H. , Handwritten notes."},{"key":"S0022481200023549_ref008","unstructured":"Magidor M. , Letter to Otmar Spinas, dated 10 30, 1988."},{"key":"S0022481200023549_ref010","unstructured":"Spinas O. , Iterated forcing in quadratic form theory (in preparation)."},{"key":"S0022481200023549_ref012","doi-asserted-by":"publisher","DOI":"10.1007\/BF02392561"},{"key":"S0022481200023549_ref004","volume-title":"Quadratic forms in infinite dimensional vector spaces","author":"Gross","year":"1979"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200023549","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,17]],"date-time":"2019-05-17T16:00:28Z","timestamp":1558108828000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200023549\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1991,12]]},"references-count":12,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1991,12]]}},"alternative-id":["S0022481200023549"],"URL":"https:\/\/doi.org\/10.2307\/2275468","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1991,12]]}}}