{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,1]],"date-time":"2026-03-01T14:29:23Z","timestamp":1772375363326,"version":"3.50.1"},"reference-count":18,"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":1197,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2010,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>An open<jats:italic>U<\/jats:italic>\u2286 \u211d is produced such that (\u211d, +, \u00b7,<jats:italic>U<\/jats:italic>) defines a Borel isomorph of (\u211d, +, \u00b7, \u2115) but does not define \u2115. It follows that (\u211d, +, \u00b7,<jats:italic>U<\/jats:italic>) defines sets in every level of the projective hierarchy but does not define all projective sets. This result is elaborated in various ways that involve geometric measure theory and working over o-minimal expansions of (\u211d, +, \u00b7). In particular, there is a Cantor set<jats:italic>E<\/jats:italic>\u2286 \u211d such that (\u211d, +, \u00b7, \u2115) defines a Borel isomorph of (\u211d, +, \u00b7, \u2115) and, for every exponentially bounded o-minimal expansion<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200002255_inline1\"\/>of (\u211d, +, \u00b7), every subset of \u211d definable in (<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200002255_inline1\"\/>,<jats:italic>E<\/jats:italic>) either has interior or is Hausdorff null.<\/jats:p>","DOI":"10.2178\/jsl\/1286198148","type":"journal-article","created":{"date-parts":[[2010,10,4]],"date-time":"2010-10-04T13:16:13Z","timestamp":1286198173000},"page":"1311-1325","source":"Crossref","is-referenced-by-count":6,"title":["Expansions of the real field by open sets: definability versus 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