{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T16:51:42Z","timestamp":1773247902036,"version":"3.50.1"},"reference-count":6,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":7954,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1992,6]]},"abstract":"<jats:p>Let <jats:bold>B<\/jats:bold> be the random real forcing. Miller [Mi] asked if there are ZFC models <jats:italic>M<\/jats:italic> \u2286 <jats:italic>N<\/jats:italic> such that forcing with <jats:bold>B<\/jats:bold><jats:sup><jats:italic>M<\/jats:italic><\/jats:sup> over <jats:italic>N<\/jats:italic> adds a dominating real. A YES answer was provided by Judah and Shelah in [JS], where in a long and sophisticated construction they built such models. In this paper we prove that forcing with <jats:bold>B<\/jats:bold><jats:sup><jats:italic>V<\/jats:italic><\/jats:sup> over <jats:bold><jats:italic>V<\/jats:italic><jats:sup>I<\/jats:sup><\/jats:bold>, where <jats:bold>I<\/jats:bold> is the infinitely often equal real forcing of [Mi], adds a dominating real over <jats:bold><jats:italic>V<\/jats:italic><jats:sup>I<\/jats:sup><\/jats:bold>. This greatly simplifies the YES answer to Miller's question. Moreover it turns out that <jats:bold>B<\/jats:bold> may be replaced here by <jats:bold>E<\/jats:bold>, the eventually different real forcing of [Mi]. This answers the second part of Miller's question. We also prove that both side by side products <jats:bold>I<\/jats:bold> \u00d7 <jats:bold>B<\/jats:bold> and <jats:bold>I<\/jats:bold> \u00d7 <jats:bold>E<\/jats:bold> add a Hechler dominating real over <jats:italic>V<\/jats:italic>.<\/jats:p><jats:p>In this section we establish the main result of the paper; namely, we prove that forcing over <jats:bold><jats:italic>V<\/jats:italic><jats:sup>I<\/jats:sup><\/jats:bold> with either of the posets <jats:bold><jats:italic>B<jats:sup>V<\/jats:sup><\/jats:italic><\/jats:bold> or <jats:bold>E<jats:sup><jats:italic>V<\/jats:italic><\/jats:sup><\/jats:bold> adds a dominating real over <jats:bold><jats:italic>V<\/jats:italic><jats:sup>I<\/jats:sup><\/jats:bold>.<\/jats:p><jats:p>First we recall the definitions of <jats:bold>I<\/jats:bold> and <jats:bold>E<\/jats:bold> from [Mi]. The infinitely often equal real forcing is the set<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200022829_eqnU1\"\/><\/jats:disp-formula><\/jats:p><jats:p>ordered by extension. Miller [Mi] proves that <jats:bold>I<\/jats:bold> is <jats:italic>\u03c9<\/jats:italic><jats:sup><jats:italic>\u03c9<\/jats:italic><\/jats:sup> bounding.<\/jats:p>","DOI":"10.2307\/2275289","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:44:45Z","timestamp":1146955485000},"page":"540-547","source":"Crossref","is-referenced-by-count":3,"title":["Adding dominating reals with <i>\u03c9<\/i><sup><i>\u03c9<\/i><\/sup> of bounding posets"],"prefix":"10.1017","volume":"57","author":[{"given":"Janusz","family":"Pawlikowski","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200022829_ref006","first-page":"595","volume-title":"Logic colloquium '76","author":"Truss","year":"1977"},{"key":"S0022481200022829_ref005","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1981-0613787-2"},{"key":"S0022481200022829_ref001","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(71)90011-8"},{"key":"S0022481200022829_ref004","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(77)90006-7"},{"key":"S0022481200022829_ref003","unstructured":"Judah H. and Shelah S. , Adding dominating reals with random algebra, preprint."},{"key":"S0022481200022829_ref002","doi-asserted-by":"publisher","DOI":"10.1090\/pspum\/013.2\/9987"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200022829","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,16]],"date-time":"2019-05-16T21:45:09Z","timestamp":1558043109000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200022829\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1992,6]]},"references-count":6,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1992,6]]}},"alternative-id":["S0022481200022829"],"URL":"https:\/\/doi.org\/10.2307\/2275289","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1992,6]]}}}