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<div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><span class="mw-redirectedfrom">(Redirected from <a href="/w/index.php?title=Azriel_Levy&redirect=no" class="mw-redirect" title="Azriel Levy">Azriel Levy</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Israeli mathematician and logician</div> <style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox biography vcard"><tbody><tr><th colspan="2" class="infobox-above" style="font-size:125%;"><div class="fn">Azriel Lévy</div></th></tr><tr><td colspan="2" class="infobox-subheader" style="font-size:125%;"><div class="nickname" lang="he">עזריאל לוי</div></td></tr><tr><th scope="row" class="infobox-label">Born</th><td class="infobox-data">c. 1934</td></tr><tr><th scope="row" class="infobox-label">Nationality</th><td class="infobox-data category">Israeli</td></tr><tr><th scope="row" class="infobox-label">Alma mater</th><td class="infobox-data">Hebrew University of Jerusalem</td></tr><tr><th scope="row" class="infobox-label">Known for</th><td class="infobox-data">Lévy hierarchy, Lévy collapse, Feferman–Levy model</td></tr><tr><td colspan="2" class="infobox-full-data"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1257001546"><b>Scientific career</b></td></tr><tr><th scope="row" class="infobox-label">Fields</th><td class="infobox-data category">Mathematics, Logic</td></tr><tr><th scope="row" class="infobox-label">Institutions</th><td class="infobox-data">Hebrew University of Jerusalem</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Doctoral_advisor" title="Doctoral advisor">Doctoral advisor</a></th><td class="infobox-data">Abraham Fraenkel, Abraham Robinson</td></tr><tr style="display:none"><td colspan="2"> </td></tr></tbody></table> <p><b>Azriel Lévy</b><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> (<a href="/wiki/Hebrew" class="mw-redirect" title="Hebrew">Hebrew</a>: עזריאל לוי; born c. 1934) is an Israeli <a href="/wiki/Mathematician" title="Mathematician">mathematician</a>, <a href="/wiki/Logician" class="mw-redirect" title="Logician">logician</a>, and a <a href="/wiki/Professor_emeritus" class="mw-redirect" title="Professor emeritus">professor emeritus</a> at the <a href="/wiki/Hebrew_University_of_Jerusalem" title="Hebrew University of Jerusalem">Hebrew University of Jerusalem</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Biography">Biography</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Azriel_L%C3%A9vy&action=edit&section=1" title="Edit section: Biography"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Lévy obtained his Ph.D. at the <a href="/wiki/Hebrew_University_of_Jerusalem" title="Hebrew University of Jerusalem">Hebrew University of Jerusalem</a> in 1958, under the supervision of <a href="/wiki/Abraham_Fraenkel" title="Abraham Fraenkel">Abraham Fraenkel</a> and <a href="/wiki/Abraham_Robinson" title="Abraham Robinson">Abraham Robinson</a>. Later, using <a href="/wiki/Paul_Cohen_(mathematician)" class="mw-redirect" title="Paul Cohen (mathematician)">Cohen</a>'s method of <a href="/wiki/Forcing_(mathematics)" title="Forcing (mathematics)">forcing</a>, he proved several results on the <a href="/wiki/Equiconsistency" title="Equiconsistency">consistency</a> of various statements contradicting the <a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a>. For example, with <a href="/wiki/Joseph_Halpern" title="Joseph Halpern">J. D. Halpern</a> he proved that the <a href="/wiki/Boolean_prime_ideal_theorem" title="Boolean prime ideal theorem">Boolean prime ideal theorem</a> does not imply the axiom of choice. He discovered the models <i>L</i>[<i>x</i>] used in <a href="/wiki/Inner_model_theory" title="Inner model theory">inner model theory</a>. He also introduced the notions of <a href="/wiki/L%C3%A9vy_hierarchy" title="Lévy hierarchy">Lévy hierarchy</a> of the formulas of <a href="/wiki/Set_theory" title="Set theory">set theory</a>, <a href="/wiki/List_of_forcing_notions#Levy_collapsing" title="List of forcing notions">Levy collapse</a> and the <a href="/w/index.php?title=Feferman%E2%80%93Levy_model&action=edit&redlink=1" class="new" title="Feferman–Levy model (page does not exist)">Feferman–Levy model</a>. </p><p>His students include <a href="/wiki/Dov_Gabbay" title="Dov Gabbay">Dov Gabbay</a>, <a href="/wiki/Moti_Gitik" title="Moti Gitik">Moti Gitik</a>, and <a href="/wiki/Menachem_Magidor" title="Menachem Magidor">Menachem Magidor</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Selected_works">Selected works</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Azriel_L%C3%A9vy&action=edit&section=2" title="Edit section: Selected works"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFLévy1958" class="citation journal cs1">Lévy, Azriel (1958). <a rel="nofollow" class="external text" href="http://matwbn.icm.edu.pl/ksiazki/fm/fm46/fm4611.pdf">"The independence of various definitions of finiteness"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/Fundamenta_Mathematicae" title="Fundamenta Mathematicae">Fundamenta Mathematicae</a></i>. <b>46</b>: 1–13. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4064%2Ffm-46-1-1-13">10.4064/fm-46-1-1-13</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Fundamenta+Mathematicae&rft.atitle=The+independence+of+various+definitions+of+finiteness&rft.volume=46&rft.pages=1-13&rft.date=1958&rft_id=info%3Adoi%2F10.4064%2Ffm-46-1-1-13&rft.aulast=L%C3%A9vy&rft.aufirst=Azriel&rft_id=http%3A%2F%2Fmatwbn.icm.edu.pl%2Fksiazki%2Ffm%2Ffm46%2Ffm4611.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAzriel+L%C3%A9vy" class="Z3988"></span></li> <li>A. Lévy: <i>A hierarchy of formulas in set theory</i>, Memoirs of the American Mathematical Society, <b>57</b>, 1965.</li> <li>J. D. Halpern, A. Lévy: The Boolean prime ideal theorem does not imply the axiom of choice, <a href="/wiki/Axiomatic_Set_Theory" class="mw-redirect" title="Axiomatic Set Theory">Axiomatic Set Theory</a>, Symposia Pure Math., 1971, 83–134.</li> <li>A. Lévy: <i>Basic Set Theory</i>, Springer-Verlag, Berlin, 1979, 391 pages;<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> reprinted by Dover Publications, 2003.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Azriel_L%C3%A9vy&action=edit&section=3" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">More commonly written with an accent in English sources, e.g., A. Lévy: <i>A hierarchy of formulas in set theory</i>, <a href="/wiki/Memoirs_of_the_American_Mathematical_Society" title="Memoirs of the American Mathematical Society">Memoirs of the American Mathematical Society</a>, <b>57</b>, 1965.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBaumgartner,_J._E.1980" class="citation journal cs1 cs1-prop-long-vol"><a href="/wiki/James_Earl_Baumgartner" title="James Earl Baumgartner">Baumgartner, J. E.</a> (1980). <a rel="nofollow" class="external text" href="https://www.ams.org/journals/bull/1980-03-01/S0273-0979-1980-14817-X/S0273-0979-1980-14817-X.pdf">"Review: <i>Basic set theory</i> by Azriel Lévy"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/Bull._Amer._Math._Soc." class="mw-redirect" title="Bull. Amer. Math. Soc.">Bull. Amer. Math. Soc.</a> (N.S.)</i>. 3, Part 1 (1): 772–774. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0273-0979-1980-14817-X">10.1090/S0273-0979-1980-14817-X</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bull.+Amer.+Math.+Soc.+%28N.S.%29&rft.atitle=Review%3A+Basic+set+theory+by+Azriel+L%C3%A9vy&rft.volume=3%2C+Part+1&rft.issue=1&rft.pages=772-774&rft.date=1980&rft_id=info%3Adoi%2F10.1090%2FS0273-0979-1980-14817-X&rft.au=Baumgartner%2C+J.+E.&rft_id=https%3A%2F%2Fwww.ams.org%2Fjournals%2Fbull%2F1980-03-01%2FS0273-0979-1980-14817-X%2FS0273-0979-1980-14817-X.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAzriel+L%C3%A9vy" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Azriel_L%C3%A9vy&action=edit&section=4" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKanamori2006" class="citation journal cs1"><a href="/wiki/Akihiro_Kanamori" title="Akihiro Kanamori">Kanamori, Akihiro</a> (2006). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.apal.2005.09.009">"Levy and set theory"</a>. <i><a href="/wiki/Annals_of_Pure_and_Applied_Logic" title="Annals of Pure and Applied Logic">Annals of Pure and Applied Logic</a></i>. <b>140</b> (1–3): 233–252. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.apal.2005.09.009">10.1016/j.apal.2005.09.009</a></span>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1089.03004">1089.03004</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Pure+and+Applied+Logic&rft.atitle=Levy+and+set+theory&rft.volume=140&rft.issue=1%E2%80%933&rft.pages=233-252&rft.date=2006&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A1089.03004%23id-name%3DZbl&rft_id=info%3Adoi%2F10.1016%2Fj.apal.2005.09.009&rft.aulast=Kanamori&rft.aufirst=Akihiro&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252Fj.apal.2005.09.009&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAzriel+L%C3%A9vy" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span 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