{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,18]],"date-time":"2026-04-18T03:39:00Z","timestamp":1776483540234,"version":"3.51.2"},"reference-count":13,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T00:00:00Z","timestamp":1772236800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T00:00:00Z","timestamp":1772236800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["2650-N35"],"award-info":[{"award-number":["2650-N35"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003977","name":"Israel Science Foundation","doi-asserted-by":"publisher","award":["1967\/21"],"award-info":[{"award-number":["1967\/21"]}],"id":[{"id":"10.13039\/501100003977","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000288","name":"Royal Society","doi-asserted-by":"publisher","award":["NF170989"],"award-info":[{"award-number":["NF170989"]}],"id":[{"id":"10.13039\/501100000288","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000265","name":"Medical Research Council","doi-asserted-by":"publisher","award":["MR\/T021705\/2"],"award-info":[{"award-number":["MR\/T021705\/2"]}],"id":[{"id":"10.13039\/501100000265","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Arch. Math. Logic"],"published-print":{"date-parts":[[2026,4]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    We continue the work from [8] and make a small\u2014but significant\u2014improvement to the definition of\n                    <jats:italic>j<\/jats:italic>\n                    -decomposable system. This provides us with a better lifting of elementary embeddings to symmetric extensions. In particular, this allows us to more easily lift weakly compact embeddings and thus preserve the notion of weakly critical cardinals. We use this improved lifting criterion to show that the first measurable cardinal can be the first weakly critical cardinal or the first Mahlo cardinal, both relative to the existence of a single measurable cardinal. However, if the first inaccessible cardinal is the first measurable cardinal, then in a suitable inner model it has Mitchell order of at least 2.\n                  <\/jats:p>","DOI":"10.1007\/s00153-026-01006-6","type":"journal-article","created":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T07:51:41Z","timestamp":1772265101000},"page":"465-475","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Small measurable cardinals"],"prefix":"10.1007","volume":"65","author":[{"given":"Yair","family":"Hayut","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Asaf","family":"Karagila","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2026,2,28]]},"reference":[{"issue":"1","key":"1006_CR1","first-page":"1","volume":"25","author":"W Arthur","year":"1983","unstructured":"Arthur, W.: Apter, Some results on consecutive large cardinals. Ann. Pure Appl. Logic 25(1), 1\u201317 (1983)","journal-title":"Ann. Pure Appl. Logic"},{"issue":"7\u20138","key":"1006_CR2","doi-asserted-by":"publisher","first-page":"863","DOI":"10.1007\/s00153-012-0302-y","volume":"51","author":"E Bilinsky","year":"2012","unstructured":"Bilinsky, E., Gitik, M.: A model with a measurable which does not carry a normal measure. Arch. Math. Logic 51(7\u20138), 863\u2013876 (2012)","journal-title":"Arch. Math. Logic"},{"issue":"2","key":"1006_CR3","doi-asserted-by":"publisher","first-page":"161","DOI":"10.1016\/0003-4843(78)90019-0","volume":"15","author":"EL Bull Jr","year":"1978","unstructured":"Bull, E.L., Jr.: Successive large cardinals. Ann. Math. Logic 15(2), 161\u2013191 (1978)","journal-title":"Ann. Math. Logic"},{"key":"1006_CR4","doi-asserted-by":"crossref","unstructured":"Cummings, J.: Iterated forcing and elementary embeddings, Handbook of set theory. Vols. 1, 2, 3, Springer, Dordrecht, pp. 775\u2013883, (2010)","DOI":"10.1007\/978-1-4020-5764-9_13"},{"key":"1006_CR5","unstructured":"Gitik, M., Hayut, Y., Karagila, A.: The first measurable can be the first inaccessible cardinal, Israel J. Math. (2024), 1\u201322, accepted for publication"},{"issue":"101","key":"1006_CR6","doi-asserted-by":"publisher","first-page":"447","DOI":"10.2307\/1970935","volume":"2","author":"S Grigorieff","year":"1975","unstructured":"Grigorieff, S.: Intermediate submodels and generic extensions in set theory. Ann. of Math. 2(101), 447\u2013490 (1975)","journal-title":"Ann. of Math."},{"issue":"3\u20134","key":"1006_CR7","doi-asserted-by":"publisher","first-page":"373","DOI":"10.1007\/s00153-015-0454-7","volume":"55","author":"Y Hayut","year":"2016","unstructured":"Hayut, Y., Karagila, A.: Restrictions on forcings that change cofinalities. Arch. Math. Logic 55(3\u20134), 373\u2013384 (2016)","journal-title":"Arch. Math. Logic"},{"issue":"1","key":"1006_CR8","doi-asserted-by":"publisher","first-page":"449","DOI":"10.1007\/s11856-020-1998-8","volume":"236","author":"Y Hayut","year":"2020","unstructured":"Hayut, Y., Karagila, A.: Critical cardinals. Israel J. Math. 236(1), 449\u2013472 (2020)","journal-title":"Israel J. Math."},{"key":"1006_CR9","unstructured":"Jech, T.: Set Theory. The Third Millennium Edition, revised and expanded, Springer Monographs in Mathematics, Springer-Verlag, Berlin, (2003)"},{"key":"1006_CR10","doi-asserted-by":"publisher","first-page":"363","DOI":"10.1007\/BF02771215","volume":"6","author":"T Jech","year":"1968","unstructured":"Jech, T.: $$\\omega _1$$ can be measurable. Israel J. Math. 6, 363\u2013367 (1968)","journal-title":"Israel J. Math."},{"issue":"1","key":"1006_CR11","doi-asserted-by":"publisher","first-page":"123","DOI":"10.1017\/jsl.2018.73","volume":"84","author":"A Karagila","year":"2019","unstructured":"Karagila, A.: Iterating symmetric extensions. J. Symb. Log. 84(1), 123\u2013159 (2019)","journal-title":"J. Symb. Log."},{"key":"1006_CR12","doi-asserted-by":"crossref","unstructured":"Mitchell, W.J.: The covering lemma, Handbook of set theory. Vols. 1, 2, 3, Springer, Dordrecht, pp. 1497\u20131594 (2010)","DOI":"10.1007\/978-1-4020-5764-9_19"},{"issue":"2","key":"1006_CR13","doi-asserted-by":"publisher","first-page":"229","DOI":"10.1017\/S030500410006151X","volume":"95","author":"JM William","year":"1984","unstructured":"William, J.M.: The core model for sequences of measures. I. Math. Proc. Cambridge Philos. Soc. 95(2), 229\u2013260 (1984)","journal-title":"I. Math. Proc. Cambridge Philos. Soc."}],"container-title":["Archive for Mathematical Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00153-026-01006-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00153-026-01006-6","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00153-026-01006-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,18]],"date-time":"2026-04-18T02:39:35Z","timestamp":1776479975000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00153-026-01006-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,2,28]]},"references-count":13,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2026,4]]}},"alternative-id":["1006"],"URL":"https:\/\/doi.org\/10.1007\/s00153-026-01006-6","relation":{},"ISSN":["0933-5846","1432-0665"],"issn-type":[{"value":"0933-5846","type":"print"},{"value":"1432-0665","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,2,28]]},"assertion":[{"value":"19 December 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 February 2026","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"28 February 2026","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}]}}