{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,6]],"date-time":"2026-04-06T15:34:11Z","timestamp":1775489651229,"version":"3.50.1"},"reference-count":15,"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":13433,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1977,6]]},"abstract":"<jats:p>Nonprincipal ultrafilters are harder to define in ZFC, and harder to obtain in ZF + DC, than nonprincipal measures.<\/jats:p><jats:p>The function \u03bc from <jats:italic>P(X)<\/jats:italic> to the closed interval [0, 1] is a measure on <jats:italic>X<\/jats:italic> if \u03bc. is finitely additive on disjoint sets and \u03bc(<jats:italic>X<\/jats:italic>) = 1. (<jats:italic>P<\/jats:italic> is the power set.) \u03bc is nonprincipal if \u03bc ({<jats:italic>x<\/jats:italic>}) = 0 for each <jats:italic>x<\/jats:italic> \u0404 <jats:italic>X<\/jats:italic>. \u03bc is an ultrafilter if Range \u03bc= {0, 1}. The existence of nonprincipal measures and ultrafilters on any infinite <jats:italic>X<\/jats:italic> follows from the axiom of choice.<\/jats:p><jats:p>Nonprincipal measures cannot necessarily be defined in ZFC. (ZF is Zermelo\u2013Fraenkel set theory. ZFC is ZF with choice.) In ZF alone they cannot even be proved to exist. This was first established by Solovay [14] using an inaccessible cardinal. In the model of [14] no nonprincipal measure on \u03c9 is even ODR (definable from ordinal and real parameters). The HODR (hereditarily ODR) sets of this model form a model of ZF + DC (dependent choice) in which no nonprincipal measure on \u03c9 exists. Pincus [8] gave a model with the same properties making no use of an inaccessible. (This model was also known to Solovay.) The second model can be combined with ideas of A. Blass [1] to give a model of ZF + DC in which no nonprincipal measures exist on any set. Using this model one obtains a model of ZFC in which no nonprincipal measure on the set of real numbers is ODR. H. Friedman, in private communication, previously obtained such a model of ZFC by a different method. Our construction will be sketched in 4.1.<\/jats:p>","DOI":"10.2307\/2272118","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:44:08Z","timestamp":1146951848000},"page":"179-190","source":"Crossref","is-referenced-by-count":25,"title":["Definability of measures and ultrafilters"],"prefix":"10.1017","volume":"42","author":[{"given":"David","family":"Pincus","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Robert M.","family":"Solovay","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200050349_ref011","volume-title":"Real and complex analysis","author":"Rudin","year":"1966"},{"key":"S0022481200050349_ref008","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0066014"},{"key":"S0022481200050349_ref005","volume-title":"Applications of model theory to algebra, analysis and probability","author":"Luxemburg","year":"1969"},{"key":"S0022481200050349_ref010","volume-title":"Real analysis","author":"Royden","year":"1968"},{"key":"S0022481200050349_ref012","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1969-0253895-6"},{"key":"S0022481200050349_ref015","first-page":"397","article-title":"Real valued measurable cardinals","volume":"XIII","author":"Solovay","journal-title":"Proceedings of Symposia in Pure Mathematics"},{"key":"S0022481200050349_ref001","unstructured":"Blass A. , A model without ultrafilters (to appear)."},{"key":"S0022481200050349_ref004","first-page":"219","article-title":"On the logical complexity of several axioms of set theory","volume":"XIII","author":"Levy","journal-title":"Proceedings of Symposia in Pure Mathematics"},{"key":"S0022481200050349_ref003","first-page":"83","article-title":"The Boolean prime ideal theorem does not imply the axiom of choice","volume":"XIII","author":"Halpern","journal-title":"Proceedings of Symposia in Pure Mathematics"},{"key":"S0022481200050349_ref002","doi-asserted-by":"publisher","DOI":"10.4064\/fm-56-3-325-345"},{"key":"S0022481200050349_ref013","doi-asserted-by":"publisher","DOI":"10.4064\/fm-30-1-96-99"},{"key":"S0022481200050349_ref006","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.28.3.108"},{"key":"S0022481200050349_ref007","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(71)90005-2"},{"key":"S0022481200050349_ref009","volume-title":"Annals of Mathematical Logic","author":"Pincus"},{"key":"S0022481200050349_ref014","doi-asserted-by":"publisher","DOI":"10.2307\/1970696"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200050349","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,27]],"date-time":"2019-05-27T21:35:57Z","timestamp":1558992957000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200050349\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1977,6]]},"references-count":15,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1977,6]]}},"alternative-id":["S0022481200050349"],"URL":"https:\/\/doi.org\/10.2307\/2272118","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1977,6]]}}}