{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,2,7]],"date-time":"2023-02-07T19:34:42Z","timestamp":1675798482752},"reference-count":21,"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":6803,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Bull. symb. log."],"published-print":{"date-parts":[[1995,6]]},"abstract":"<jats:p>It was questions about points on the real line that initiated the study of set theory. Points paved the way to point sets and these to ever more abstract sets. And there was more: Reflection on structural properties of point sets not only <jats:italic>initiated<\/jats:italic> the study of ordinary sets; it also supplied blueprints for defining extra-ordinary, \u201clarge\u201d sets, transcending those provided by standard set theory. In return, the existence of such large sets turned out critical to settling open conjectures about point sets.<\/jats:p><jats:p>How to explain such action at a distance between the very large and the rather small? Rather than having an air of magic, could these results rest on deep structural similarities between the two superficially distant species of sets?<\/jats:p><jats:p>In this essay I dissect one group of such two-way results. Their linchpin is the notion of <jats:italic>measure<\/jats:italic>.<\/jats:p><jats:p>\u00a7<jats:bold>1. Vitali's impossibility result<\/jats:bold>. Our starting point is a problem in measure theory regarding the notion of \u201cLebesgue measure.\u201d Before presenting the problem, I would like to review the notion of Lebesgue measure. Rather than listing its main properties, I would like to show how Lebesgue measure is born out of an attempt to generalize the notion of the length of an interval to arbitrary sets of reals. One tries to approximate arbitrary sets of reals by intervals, in the hope that the lengths of the intervals will induce a measure on these sets.<\/jats:p>","DOI":"10.2307\/421039","type":"journal-article","created":{"date-parts":[[2006,5,7]],"date-time":"2006-05-07T07:07:50Z","timestamp":1146985670000},"page":"170-188","source":"Crossref","is-referenced-by-count":2,"title":["Measures: Back and Forth Between Point sets and Large sets"],"prefix":"10.1017","volume":"1","author":[{"given":"Noa","family":"Goldring","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,1,15]]},"reference":[{"key":"S1079898600008210_ref004","volume-title":"Collected works, Volume II: Publications 1938\u20131974","author":"G\u00f6del","year":"1990"},{"key":"S1079898600008210_ref021","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.85.18.6587"},{"key":"S1079898600008210_ref010","doi-asserted-by":"publisher","DOI":"10.1007\/BF02760523"},{"key":"S1079898600008210_ref016","doi-asserted-by":"publisher","DOI":"10.2307\/1970696"},{"key":"S1079898600008210_ref019","doi-asserted-by":"crossref","first-page":"140","DOI":"10.4064\/fm-16-1-140-150","article-title":"Zur Masstheorie in der allgemeinen Mengenlehre","volume":"16","author":"Ulam","year":"1930","journal-title":"Fundamenta Mathematica"},{"key":"S1079898600008210_ref006","first-page":"91","article-title":"Sur la classification de M. Baire","volume":"164","author":"Lusin","year":"1917","journal-title":"Comptes Rendus de l'Academie des Sciences, Paris"},{"key":"S1079898600008210_ref018","unstructured":"Solovay R. , G\u00f6del 1938: Introductory note to 1938, 1939, 1939a and 1940, in G\u00f6del [5]."},{"key":"S1079898600008210_ref011","volume-title":"Real analysis","author":"Royden","year":"1963"},{"key":"S1079898600008210_ref001","volume-title":"Set theory","author":"Drake","year":"1974"},{"key":"S1079898600008210_ref012","doi-asserted-by":"publisher","DOI":"10.1007\/BF02760522"},{"key":"S1079898600008210_ref020","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511609596"},{"key":"S1079898600008210_ref009","volume-title":"Descriptive set theory","author":"Moschovakis","year":"1980"},{"key":"S1079898600008210_ref002","volume-title":"Real analysis","author":"Folland","year":"1984"},{"key":"S1079898600008210_ref013","first-page":"381","article-title":"Large cardinals and Lebesgue measure","author":"Shelah","year":"1980","journal-title":"Israel Journal of Mathematics"},{"key":"S1079898600008210_ref008","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1090\/S0894-0347-1989-0955605-X","article-title":"Projective determinacy","volume":"2","author":"Margin","year":"1989","journal-title":"Journal of the American Mathematical Society"},{"key":"S1079898600008210_ref015","first-page":"59","volume-title":"Symposium papers commemorating the sixtieth birthday of Kurt G\u00f6del","author":"Solovay","year":"1969"},{"key":"S1079898600008210_ref014","volume-title":"Essays on the foundations of mathematics","author":"Shoenfield","year":"1961"},{"key":"S1079898600008210_ref003","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.24.12.556"},{"key":"S1079898600008210_ref007","first-page":"35","volume-title":"Bulletin International de l'Academie des Sciences de Cracovie","author":"Lusin","year":"1918"},{"key":"S1079898600008210_ref005","volume-title":"Set theory","author":"Jech","year":"1978"},{"key":"S1079898600008210_ref017","doi-asserted-by":"publisher","DOI":"10.1090\/pspum\/013.1\/0290961"}],"container-title":["Bulletin of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S1079898600008210","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,13]],"date-time":"2019-05-13T21:48:33Z","timestamp":1557784113000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S1079898600008210\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1995,6]]},"references-count":21,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1995,6]]}},"alternative-id":["S1079898600008210"],"URL":"https:\/\/doi.org\/10.2307\/421039","relation":{},"ISSN":["1079-8986","1943-5894"],"issn-type":[{"value":"1079-8986","type":"print"},{"value":"1943-5894","type":"electronic"}],"subject":[],"published":{"date-parts":[[1995,6]]}}}