{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,10,6]],"date-time":"2023-10-06T14:14:48Z","timestamp":1696601688050},"reference-count":21,"publisher":"Wiley","issue":"1-2","license":[{"start":{"date-parts":[[2014,2,3]],"date-time":"2014-02-03T00:00:00Z","timestamp":1391385600000},"content-version":"vor","delay-in-days":2,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematical Logic Qtrly"],"published-print":{"date-parts":[[2014,2]]},"abstract":"<jats:p>The von Neumann hierarchy of sets is heavily used as a basic tool in classical set theory, being an underlying ingredient in many proofs and concepts. In constructive set theories like <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201200100-math-0001.png\" xlink:title=\"urn:x-wiley:09425616:malq201200100:equation:malq201200100-math-0001\" \/> without the powerset axiom however, it loses much of its potency by ceasing to be a hierarchy of sets as its single stages become only classes. This article proposes an alternative cumulative hierarchy which does not have this drawback and provides examples of how it can be used to prove new theorems in <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201200100-math-0002.png\" xlink:title=\"urn:x-wiley:09425616:malq201200100:equation:malq201200100-math-0002\" \/>.<\/jats:p>","DOI":"10.1002\/malq.201200100","type":"journal-article","created":{"date-parts":[[2014,2,4]],"date-time":"2014-02-04T15:25:45Z","timestamp":1391527545000},"page":"21-30","source":"Crossref","is-referenced-by-count":0,"title":["A cumulative hierarchy of sets for constructive set theory"],"prefix":"10.1002","volume":"60","author":[{"given":"Albert","family":"Ziegler","sequence":"first","affiliation":[{"name":"School of Mathematics University of Leeds  Leeds LS2 9JT United Kingdom"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2014,2,3]]},"reference":[{"key":"e_1_2_11_2_1","first-page":"55","volume-title":"Studies in Logic and the Foundations of Mathematics","author":"Aczel P.","year":"1978"},{"key":"e_1_2_11_3_1","doi-asserted-by":"publisher","DOI":"10.1002\/malq.200710035"},{"key":"e_1_2_11_4_1","unstructured":"P.Aczel Explicit set existence talk at Leeds Symposium on Proof Theory and Constructivism 3\u201316 July2009 unpublished."},{"key":"e_1_2_11_5_1","volume-title":"preprint","author":"Aczel P.","year":"2001"},{"key":"e_1_2_11_6_1","unstructured":"A.Bauer First steps in synthetic computability theory talk at Fischbachau Germany 18 September2005 unpublished."},{"key":"e_1_2_11_7_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.apal.2005.05.021"},{"key":"e_1_2_11_8_1","unstructured":"B.Gibbons M.RathjenandK.Thiel Relating Mahlo to Veblen via realizability and ordinal analysis manuscript (2005)."},{"key":"e_1_2_11_9_1","volume-title":"Springer Monographs in Mathematics","author":"Kanamori A.","year":"2009"},{"key":"e_1_2_11_10_1","volume-title":"Studies in Logic and the Foundations of Mathematics","author":"Kunen K.","year":"1980"},{"key":"e_1_2_11_11_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.apal.2004.08.002"},{"key":"e_1_2_11_12_1","volume-title":"Studies in Proof Theory","author":"Martin\u2010L\u00f6f P.","year":"1984"},{"key":"e_1_2_11_13_1","doi-asserted-by":"publisher","DOI":"10.2307\/2272159"},{"key":"e_1_2_11_14_1","article-title":"The strength of some Martin\u2010L\u00f6f type theories","author":"Rathjen M.","year":"1993","journal-title":"preprint"},{"key":"e_1_2_11_15_1","first-page":"275","volume-title":"Lecture Notes in Logic","author":"Rathjen M.","year":"1998"},{"key":"e_1_2_11_16_1","first-page":"442","volume-title":"Lecture Notes in Logic","author":"Rathjen M.","year":"2006"},{"key":"e_1_2_11_17_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0168-0072(97)00072-9"},{"key":"e_1_2_11_18_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.apal.2005.12.008"},{"key":"e_1_2_11_19_1","unstructured":"A.Swan CZF does not have the Existence Property preprint 2013 (arXiv:1303.3408)."},{"key":"e_1_2_11_20_1","volume-title":"Ph.D. Thesis","author":"Berg B. van\u00a0den","year":"2006"},{"key":"e_1_2_11_21_1","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1515\/crll.1929.160.227","article-title":"\u00dcber eine Widerspruchsfreiheitsfrage in der axiomatischen Mengenlehre","volume":"160","author":"Neumann J. von","year":"1929","journal-title":"J. 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