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Philosophical background: iteration, ineffability, reflection.<\/jats:bold> There are at least two heuristic motivations for the axioms of standard set theory, by which we mean, as usual, first-order Zermelo\u2013Fraenkel set theory with the axiom of choice (ZFC): the iterative conception and limitation of size (see Boolos, 1989). Each strand provides a rather hospitable environment for the hypothesis that the set-theoretic universe is ineffable, which is our target in this paper, although the motivation is different in each case.<\/jats:p>","DOI":"10.1017\/s1755020308080192","type":"journal-article","created":{"date-parts":[[2008,9,10]],"date-time":"2008-09-10T08:18:40Z","timestamp":1221034720000},"page":"241-266","source":"Crossref","is-referenced-by-count":1,"title":["FREGE MEETS ZERMELO: A PERSPECTIVE ON INEFFABILITY AND REFLECTION"],"prefix":"10.1017","volume":"1","author":[{"given":"STEWART","family":"SHAPIRO","sequence":"first","affiliation":[]},{"given":"GABRIEL","family":"UZQUIANO","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2008,8,1]]},"reference":[{"key":"S1755020308080192_manual_ref-7","doi-asserted-by":"publisher","DOI":"10.1093\/aristoteliansupp\/67.1.213"},{"key":"S1755020308080192_manual_ref-18","doi-asserted-by":"publisher","DOI":"10.1023\/B:LOGI.0000046129.22633.96"},{"key":"S1755020308080192_manual_ref-9","first-page":"257","volume-title":"Between Logic and Intuition: Essays in Honor of Charles Parsons","author":"Boolos","year":"1998"},{"key":"S1755020308080192_manual_ref-24","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4613-9478-5"},{"key":"S1755020308080192_manual_ref-33","doi-asserted-by":"publisher","DOI":"10.2307\/421203"},{"key":"S1755020308080192_manual_ref-35","first-page":"339","article-title":"Die Axiomatizierung der Mengenlehre","volume":"27","author":"von Neumann","year":"1928","journal-title":"Mathematische Zeitschrift"},{"key":"S1755020308080192_manual_ref-34","doi-asserted-by":"publisher","DOI":"10.1515\/crll.1925.154.219"},{"key":"S1755020308080192_manual_ref-21","doi-asserted-by":"publisher","DOI":"10.4064\/fm-49-1-1-10"},{"key":"S1755020308080192_manual_ref-3","first-page":"3","volume-title":"Essays on the Foundations of Mathematics","author":"Bernays","year":"1961"},{"key":"S1755020308080192_manual_ref-29","doi-asserted-by":"publisher","DOI":"10.1093\/oso\/9780199276424.003.0010"},{"volume-title":"Die Grundlagen der theoretischen Logik","year":"1928","author":"Hilbert","key":"S1755020308080192_manual_ref-17"},{"volume-title":"Foundations without Foundationalism: A Case for Second-Order Logic","year":"1991","author":"Shapiro","key":"S1755020308080192_manual_ref-27"},{"key":"S1755020308080192_manual_ref-30","first-page":"65","article-title":"The iterative hierarchy of sets","volume":"39","author":"Tait","year":"1990","journal-title":"Ivyum"},{"volume-title":"From Mathematics to Philosophy","year":"1974","author":"Wang","key":"S1755020308080192_manual_ref-36"},{"key":"S1755020308080192_manual_ref-15","doi-asserted-by":"publisher","DOI":"10.1305\/ndjfl\/1038336882"},{"key":"S1755020308080192_manual_ref-4","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(08)70893-0"},{"volume-title":"Frege: Philosophy of Mathematics","year":"1991","author":"Dummett","key":"S1755020308080192_manual_ref-12"},{"key":"S1755020308080192_manual_ref-31","doi-asserted-by":"publisher","DOI":"10.1093\/oso\/9780198236542.003.0019"},{"volume-title":"Similar Subclasses","year":"2003","author":"Friedman","key":"S1755020308080192_manual_ref-14"},{"key":"S1755020308080192_manual_ref-25","doi-asserted-by":"publisher","DOI":"10.1093\/oso\/9780199276424.001.0001"},{"key":"S1755020308080192_manual_ref-5","doi-asserted-by":"publisher","DOI":"10.2307\/2025204"},{"key":"S1755020308080192_manual_ref-16","doi-asserted-by":"publisher","DOI":"10.1093\/0198236395.001.0001"},{"volume-title":"Foundations of Set Theory","year":"1973","author":"Bar-Hillel","key":"S1755020308080192_manual_ref-2"},{"key":"S1755020308080192_manual_ref-13","doi-asserted-by":"publisher","DOI":"10.1080\/0144534031000124134"},{"key":"S1755020308080192_manual_ref-37","doi-asserted-by":"publisher","DOI":"10.4064\/fm-16-1-29-47"},{"key":"S1755020308080192_manual_ref-10","doi-asserted-by":"publisher","DOI":"10.1093\/philmat\/12.3.193"},{"key":"S1755020308080192_manual_ref-1","doi-asserted-by":"publisher","DOI":"10.1305\/ndjfl\/1107220671"},{"key":"S1755020308080192_manual_ref-6","doi-asserted-by":"publisher","DOI":"10.5840\/philtopics19891721"},{"key":"S1755020308080192_manual_ref-22","doi-asserted-by":"publisher","DOI":"10.2307\/2315201"},{"key":"S1755020308080192_manual_ref-11","doi-asserted-by":"publisher","DOI":"10.1515\/9780691187068"},{"volume-title":"Logic, logic, and logic","year":"1998","author":"Boolos","key":"S1755020308080192_manual_ref-8"},{"key":"S1755020308080192_manual_ref-26","doi-asserted-by":"publisher","DOI":"10.1007\/BF00152934"},{"key":"S1755020308080192_manual_ref-19","unstructured":"Koellner, P . 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