{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T04:27:24Z","timestamp":1775795244602,"version":"3.50.1"},"reference-count":12,"publisher":"World Scientific Pub Co Pte Ltd","issue":"02","funder":[{"name":"DFG, German Research Foundation","award":["EXC 2044-390685587"],"award-info":[{"award-number":["EXC 2044-390685587"]}]},{"name":"the Austrian Science Fund","award":["10.55776\/Y1498"],"award-info":[{"award-number":["10.55776\/Y1498"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Math. Log."],"published-print":{"date-parts":[[2025,8]]},"abstract":"<jats:p> According to a theorem due to Kenneth Kunen, under ZFC, there is no ordinal [Formula: see text] and nontrivial elementary embedding [Formula: see text]. His proof relied on the Axiom of Choice (AC), and no proof from ZF alone is has been discovered. <\/jats:p><jats:p> [Formula: see text] is the assertion, introduced by Hugh Woodin, that [Formula: see text] is an ordinal and there is an elementary embedding [Formula: see text] with critical point [Formula: see text]. And [Formula: see text] asserts that [Formula: see text] holds for some [Formula: see text]. The axiom [Formula: see text] is one of the strongest large cardinals not known to be inconsistent with AC.\u00a0It is usually studied assuming ZFC in the full universe V (in which case [Formula: see text] must be a limit ordinal), but we assume only ZF. <\/jats:p><jats:p> We prove, assuming [Formula: see text] \u201c[Formula: see text] is an even ordinal\u201d, that there is a proper class transitive inner model M containing [Formula: see text] and satisfying [Formula: see text] \u201cthere is an elementary embedding [Formula: see text]\u201d; in fact we will have [Formula: see text], where j witnesses [Formula: see text] in M. This result was first proved by the author under the added assumption that [Formula: see text] exists; Gabe Goldberg noticed that this extra assumption was unnecessary. If also [Formula: see text] is a limit ordinal and [Formula: see text] holds in V, then the model M will also satisfy [Formula: see text]. <\/jats:p><jats:p> We show that [Formula: see text] \u201c[Formula: see text] is even\u201d [Formula: see text] implies [Formula: see text] exists for every [Formula: see text], but if consistent, this theory does not imply [Formula: see text] exists. <\/jats:p>","DOI":"10.1142\/s0219061324500132","type":"journal-article","created":{"date-parts":[[2024,2,14]],"date-time":"2024-02-14T04:39:02Z","timestamp":1707885542000},"source":"Crossref","is-referenced-by-count":1,"title":["On the consistency of ZF with an elementary embedding from V\u03bb+2 into V\u03bb+2"],"prefix":"10.1142","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3197-8197","authenticated-orcid":false,"given":"Farmer","family":"Schlutzenberg","sequence":"first","affiliation":[{"name":"Institut f\u00fcr Diskrete Mathematik und Geometrie, Technische Universit\u00e4t Wien, Wiedner Hauptstra\u00dfe 8\u201310\/104, 1040 Wien, Austria"}]}],"member":"219","published-online":{"date-parts":[[2024,3,16]]},"reference":[{"key":"S0219061324500132BIB002","doi-asserted-by":"publisher","DOI":"10.1090\/pspum\/013.2\/0401475"},{"key":"S0219061324500132BIB003","doi-asserted-by":"publisher","DOI":"10.2307\/2269948"},{"key":"S0219061324500132BIB004","doi-asserted-by":"publisher","DOI":"10.1142\/S0219061315500014"},{"key":"S0219061324500132BIB005","doi-asserted-by":"publisher","DOI":"10.1007\/s40574-017-0136-y"},{"key":"S0219061324500132BIB006","doi-asserted-by":"publisher","DOI":"10.1142\/9789814678001_0012"},{"key":"S0219061324500132BIB007","doi-asserted-by":"publisher","DOI":"10.1142\/S021906131100102X"},{"key":"S0219061324500132BIB010","doi-asserted-by":"crossref","unstructured":"J. R. Steel and  W. Hugh Woodin,  HOD as a Core Model; Ordinal Definability and Recursion Theory: The Cabal Seminar, Vol.  3  (Cambridge Books Online,  2016), pp. 257\u2013346.","DOI":"10.1017\/CBO9781139519694.010"},{"key":"S0219061324500132BIB012","doi-asserted-by":"publisher","DOI":"10.1017\/jsl.2018.5"},{"key":"S0219061324500132BIB013","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4020-5764-9_20"},{"key":"S0219061324500132BIB014","doi-asserted-by":"publisher","DOI":"10.2307\/2586799"},{"key":"S0219061324500132BIB016","series-title":"Springer Monographs in Mathematics","volume-title":"The Higher Infinite: Large Cardinals in Set Theory from their Beginnings","author":"Kanamori A.","year":"2005","edition":"2"},{"key":"S0219061324500132BIB017","doi-asserted-by":"publisher","DOI":"10.1016\/S0168-0072(97)00031-6"}],"container-title":["Journal of Mathematical Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0219061324500132","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,4,10]],"date-time":"2025-04-10T07:06:05Z","timestamp":1744268765000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/10.1142\/S0219061324500132"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,3,16]]},"references-count":12,"journal-issue":{"issue":"02","published-print":{"date-parts":[[2025,8]]}},"alternative-id":["10.1142\/S0219061324500132"],"URL":"https:\/\/doi.org\/10.1142\/s0219061324500132","relation":{},"ISSN":["0219-0613","1793-6691"],"issn-type":[{"value":"0219-0613","type":"print"},{"value":"1793-6691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,3,16]]},"article-number":"2450013"}}