{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,13]],"date-time":"2026-03-13T11:26:22Z","timestamp":1773401182422,"version":"3.50.1"},"reference-count":15,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2014,1,15]],"date-time":"2014-01-15T00:00:00Z","timestamp":1389744000000},"content-version":"unspecified","delay-in-days":4976,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Bull. symb. log."],"published-print":{"date-parts":[[2000,6]]},"abstract":"<jats:p>The present paper investigates the power of proper forcings to change the shape of the universe, in a certain well-defined respect. It turns out that the ranking among large cardinals can be used as a measure for that power. However, in order to establish the final result I had to isolate a new large cardinal concept, which I dubbed \u201cremarkability.\u201d Let us approach the exact formulation of the problem\u2014and of its solution\u2014at a slow pace.<\/jats:p><jats:p>Breathtaking developments in the mid 1980s found one of its culminations in the theorem, due to Martin, Steel, and Woodin, that the existence of infinitely many Woodin cardinals with a measurable cardinal above them all implies that AD, the axiom of determinacy, holds in the least inner model containing all the reals, L(\u211d) (cf. [6[, p. 91). One of the nice things about AD is that the theory <jats:italic>ZF + AD + V = L<\/jats:italic>(\u211d) appears as a choiceless \u201ccompletion\u201d of ZF in that any interesting question (in particular, about sets of reals) seems to find an at least attractive answer in that theory (cf., for example, [5] Chap. 6). (Compare with ZF + V = L!) Beyond that, AD is very canonical as may be illustrated as follows.<\/jats:p><jats:p>Let us say that L(\u211d) is absolute for set-sized forcings if for all posets P \u2208 V, for all formulae \u03d5, and for all <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1079898600006569_inline2\"\/> \u2208 \u211d do we have that<\/jats:p><jats:p><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1079898600006569_inline1\"\/><\/jats:p><jats:p>where <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1079898600006569_inline3\"\/> is a name for the set of reals in the extension.<\/jats:p>","DOI":"10.2307\/421205","type":"journal-article","created":{"date-parts":[[2006,5,7]],"date-time":"2006-05-07T03:13:57Z","timestamp":1146971637000},"page":"176-184","source":"Crossref","is-referenced-by-count":20,"title":["Proper Forcing and Remarkable Cardinals"],"prefix":"10.1017","volume":"6","author":[{"given":"Ralf-Dieter","family":"Schindler","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,1,15]]},"reference":[{"key":"S1079898600006569_ref006","doi-asserted-by":"publisher","DOI":"10.1090\/S0894-0347-1989-0955605-X"},{"key":"S1079898600006569_ref013","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(93)90037-E"},{"key":"S1079898600006569_ref011","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-21543-2"},{"key":"S1079898600006569_ref009","volume-title":"Transactions of the American Mathematical Society","author":"Schindler"},{"key":"S1079898600006569_ref003","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(94)00031-W"},{"key":"S1079898600006569_ref010","article-title":"Proper forcing and remarkable cardinals II","author":"Neeman","journal-title":"Journal of Symbolic Logic"},{"key":"S1079898600006569_ref012","unstructured":"Steel J. , Core models with more Woodin cardinals, preprint."},{"key":"S1079898600006569_ref001","doi-asserted-by":"publisher","DOI":"10.2307\/2275649"},{"key":"S1079898600006569_ref005","volume-title":"The higher infinite","author":"Kanamori","year":"1994"},{"key":"S1079898600006569_ref004","first-page":"84","volume-title":"Mathematical logic and foundations of set theory","author":"Jensen","year":"1970"},{"key":"S1079898600006569_ref007","unstructured":"Neeman I. and Zapletal J. , Proper forcing and L(\u211d), preprint."},{"key":"S1079898600006569_ref014","unstructured":"Woodin H. , Lecture in the spring of 1990, notes taken by Burke D. and Schimmerling E. ."},{"key":"S1079898600006569_ref002","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511629198"},{"key":"S1079898600006569_ref008","first-page":"281","article-title":"Proper forcing and absoluteness in L(\u211d)","volume":"39","author":"Neeman","year":"1998","journal-title":"Commentationes Mathematicae Universitatis Carolinae"},{"key":"S1079898600006569_ref015","doi-asserted-by":"publisher","DOI":"10.1515\/9783110804737"}],"container-title":["Bulletin of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S1079898600006569","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,8]],"date-time":"2019-05-08T17:17:14Z","timestamp":1557335834000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S1079898600006569\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2000,6]]},"references-count":15,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2000,6]]}},"alternative-id":["S1079898600006569"],"URL":"https:\/\/doi.org\/10.2307\/421205","relation":{},"ISSN":["1079-8986","1943-5894"],"issn-type":[{"value":"1079-8986","type":"print"},{"value":"1943-5894","type":"electronic"}],"subject":[],"published":{"date-parts":[[2000,6]]}}}