{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,1]],"date-time":"2026-03-01T13:45:36Z","timestamp":1772372736045,"version":"3.50.1"},"reference-count":16,"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":376,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2013,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We investigate the question \u201cTo what extent can random reals be used as a tool to establish number theoretic facts?\u201d Let 2-<jats:italic>RAN<\/jats:italic>be the principle that for every real<jats:italic>X<\/jats:italic>there is a real<jats:italic>R<\/jats:italic>which is 2-random relative to<jats:italic>X<\/jats:italic>. In Section 2, we observe that the arguments of Csima and Mileti [3] can be implemented in the base theory<jats:italic>RCA<\/jats:italic><jats:sub>0<\/jats:sub>and so<jats:italic>RCA<\/jats:italic><jats:sub>0<\/jats:sub>+ 2-<jats:italic>RAN<\/jats:italic>implies the Rainbow Ramsey Theorem. In Section 3, we show that the Rainbow Ramsey Theorem is not conservative over<jats:italic>RCA<\/jats:italic><jats:sub>0<\/jats:sub>for arithmetic sentences. Thus, from the Csima\u2013Mileti fact that the existence of random reals has infinitary-combinatorial consequences we can conclude that 2-<jats:italic>RAN<\/jats:italic>has non-trivial arithmetic consequences. In Section 4, we show that 2-<jats:italic>RAN<\/jats:italic>is conservative over<jats:italic>RCA<\/jats:italic><jats:sub>0<\/jats:sub>+<jats:italic>B<\/jats:italic>\u03a3<jats:sub>2<\/jats:sub>for<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200000359_inline01\"\/>-sentences. Thus, the set of first-order consequences of 2-<jats:italic>RAN<\/jats:italic>is strictly stronger than<jats:italic>P<\/jats:italic><jats:sup>\u2212<\/jats:sup>+<jats:italic>I<\/jats:italic>\u03a3<jats:sub>1<\/jats:sub>and no stronger than<jats:italic>P<\/jats:italic><jats:sup>\u2212<\/jats:sup>+<jats:italic>B<\/jats:italic>\u03a3<jats:sub>2<\/jats:sub>.<\/jats:p>","DOI":"10.2178\/jsl.7801130","type":"journal-article","created":{"date-parts":[[2013,1,23]],"date-time":"2013-01-23T14:38:23Z","timestamp":1358951903000},"page":"195-206","source":"Crossref","is-referenced-by-count":7,"title":["Random reals, the rainbow Ramsey theorem, and arithmetic conservation"],"prefix":"10.1017","volume":"78","author":[{"given":"Chris J.","family":"Conidis","sequence":"first","affiliation":[]},{"given":"Theodore A.","family":"Slaman","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200000359_ref002","first-page":"1","volume":"66","author":"Cholak","year":"2001","journal-title":"On the strength of Ramsey's theorem for pairs"},{"key":"S0022481200000359_ref010","doi-asserted-by":"publisher","DOI":"10.1112\/S002461079600470X"},{"key":"S0022481200000359_ref016","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-02460-7"},{"key":"S0022481200000359_ref006","unstructured":"Hirst J. 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