{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,21]],"date-time":"2026-01-21T04:15:34Z","timestamp":1768968934113,"version":"3.49.0"},"reference-count":28,"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":5945,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1997,12]]},"abstract":"<jats:p>Some of the most striking results in modern set theory have emerged from the study of simply-definable sets of real numbers. Indeed, simple questions like: what are the posible cardinalities?, are they measurable?, do they have the property of Baire?, etc., cannot be answered in ZFC.<\/jats:p><jats:p>When one restricts the attention to the analytic sets, i.e., the continuous images of Borel sets, then ZFC does provide an answer to these questions. But this is no longer true for the projective sets, i.e., all the sets of reals that can be obtained from the Borel sets by taking continuous images and complements. In this paper we shall concentrate on particular projective classes, the <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200015814_inline1\"\/>, and using forcing constructions we will produce models of ZFC where, for some <jats:italic>n<\/jats:italic>, all <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200015814_inline1\"\/>, sets have some specified property. For the definition and basic facts about the projective classes <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200015814_inline2\"\/>, and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200015814_inline1\"\/>, as well as the Kleene (or lightface) classes <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200015814_inline3\"\/>, and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200015814_inline4\"\/>, we refer the reader to Moschovakis [19].<\/jats:p><jats:p>The first part of the paper is about measure and category. Early in this century, Luzin [16] and Luzin-Sierpi\u0144ski [17] showed that all analytic (i.e., <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200015814_inline5\"\/>) sets of reals are Lebesgue measurable and have the property of Baire.<\/jats:p>","DOI":"10.2307\/2275649","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T23:03:06Z","timestamp":1146956586000},"page":"1379-1428","source":"Crossref","is-referenced-by-count":11,"title":["sets of reals"],"prefix":"10.1017","volume":"62","author":[{"given":"Joan","family":"Bagaria","sequence":"first","affiliation":[]},{"given":"W. Hugh","family":"Woodin","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200015814_ref026","first-page":"595","volume-title":"Logic colloquium '76","author":"Truss"},{"key":"S0022481200015814_ref025","doi-asserted-by":"publisher","DOI":"10.2307\/1970860"},{"key":"S0022481200015814_ref022","doi-asserted-by":"publisher","DOI":"10.4064\/fm-103-1-47-60"},{"key":"S0022481200015814_ref020","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4615-9964-7"},{"key":"S0022481200015814_ref018","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(70)90009-4"},{"key":"S0022481200015814_ref017","first-page":"53","article-title":"Sur un ensemble non measurable B","volume":"2","author":"Luzin","year":"1923","journal-title":"Journal de Math\u00e9matiques. Neuvi\u00e9me S\u00e9rie"},{"key":"S0022481200015814_ref016","first-page":"91","article-title":"Sur la classification de M. Baire","volume":"164","author":"Luzin","year":"1917","journal-title":"Comptes Rendus de l'Acad\u00e9mie des Sciences Paris"},{"key":"S0022481200015814_ref014","volume-title":"Handbook of Boolean algebras","author":"Koppelberg","year":"1989"},{"key":"S0022481200015814_ref010","doi-asserted-by":"publisher","DOI":"10.1305\/ndjfl\/1093870823"},{"key":"S0022481200015814_ref009","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(77)90004-3"},{"key":"S0022481200015814_ref011","volume-title":"Set theory","author":"Jech","year":"1978"},{"key":"S0022481200015814_ref008","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4684-9440-2"},{"key":"S0022481200015814_ref007","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.24.12.556"},{"key":"S0022481200015814_ref004","doi-asserted-by":"publisher","DOI":"10.2307\/1999530"},{"key":"S0022481200015814_ref003","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(94)90017-5"},{"key":"S0022481200015814_ref002","unstructured":"Bagaria J. , The preservation of forcing axioms under forcing with measure algebras, unpublished, 1992."},{"key":"S0022481200015814_ref028","unstructured":"Woodin W. H. , Discontinuous homomorphisms of C (\u03a9) and set theory, Ph.D. dissertation , University of California, Berkeley, 1984."},{"key":"S0022481200015814_ref013","first-page":"72","volume":"58","author":"Judah","year":"1993","journal-title":"sets of reals"},{"key":"S0022481200015814_ref023","doi-asserted-by":"publisher","DOI":"10.1007\/BF02760522"},{"key":"S0022481200015814_ref001","unstructured":"Bagaria J. , Definable forcing and regularity properties of projective sets of reals, Ph.D. thesis , Berkeley, 1991."},{"key":"S0022481200015814_ref006","volume-title":"Handbook of Boolean algebras","author":"Fremlin","year":"1989"},{"key":"S0022481200015814_ref005","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(82)90002-X"},{"key":"S0022481200015814_ref012","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1016\/0168-0072(89)90016-X","article-title":"sets of reals","volume":"42","author":"Judah","year":"1989","journal-title":"Annals of Pure and Applied Logic"},{"key":"S0022481200015814_ref021","doi-asserted-by":"publisher","DOI":"10.1007\/BF02759764"},{"key":"S0022481200015814_ref015","unstructured":"Kunen K. , (\u03ba, \u03bb*)-gaps under MA, handwritten notes, 1976."},{"key":"S0022481200015814_ref019","volume-title":"Descriptive set theory","author":"Moschovakis","year":"1980"},{"key":"S0022481200015814_ref027","first-page":"365","volume-title":"Logic colloquium '81","author":"Woodin"},{"key":"S0022481200015814_ref024","first-page":"1","volume-title":"Annals of Mathematics","author":"Solovay","year":"1970"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200015814","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,11]],"date-time":"2019-05-11T19:55:09Z","timestamp":1557604509000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200015814\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1997,12]]},"references-count":28,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1997,12]]}},"alternative-id":["S0022481200015814"],"URL":"https:\/\/doi.org\/10.2307\/2275649","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1997,12]]}}}