{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T16:27:21Z","timestamp":1776702441628,"version":"3.51.2"},"reference-count":16,"publisher":"World Scientific Pub Co Pte Lt","issue":"02","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Math. Log."],"published-print":{"date-parts":[[2017,12]]},"abstract":"<jats:p> A transitive model [Formula: see text] of ZFC is called a ground if the universe [Formula: see text] is a set forcing extension of [Formula: see text]. We show that the grounds of[Formula: see text][Formula: see text][Formula: see text] are downward set-directed. Consequently, we establish some fundamental theorems on the forcing method and the set-theoretic geology. For instance, (1) the mantle, the intersection of all grounds, must be a model of ZFC. (2) [Formula: see text] has only set many grounds if and only if the mantle is a ground. We also show that if the universe has some very large cardinal, then the mantle must be a ground. <\/jats:p>","DOI":"10.1142\/s021906131750009x","type":"journal-article","created":{"date-parts":[[2017,9,12]],"date-time":"2017-09-12T03:33:52Z","timestamp":1505187232000},"page":"1750009","source":"Crossref","is-referenced-by-count":31,"title":["The downward directed grounds hypothesis and very large cardinals"],"prefix":"10.1142","volume":"17","author":[{"given":"Toshimichi","family":"Usuba","sequence":"first","affiliation":[{"name":"Faculty of Science and Engineering, Waseda University, Okubo 3-4-1, Shinjyuku, Tokyo 169-8555, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2017,11,27]]},"reference":[{"key":"S021906131750009XBIB001","doi-asserted-by":"publisher","DOI":"10.1007\/s00153-015-0458-3"},{"key":"S021906131750009XBIB002","doi-asserted-by":"publisher","DOI":"10.4064\/fm-83-1-35-46"},{"key":"S021906131750009XBIB003","doi-asserted-by":"publisher","DOI":"10.1142\/9789813223523_0003"},{"key":"S021906131750009XBIB004","doi-asserted-by":"publisher","DOI":"10.1016\/j.apal.2014.11.004"},{"key":"S021906131750009XBIB005","doi-asserted-by":"publisher","DOI":"10.4064\/fm180-3-4"},{"key":"S021906131750009XBIB006","doi-asserted-by":"publisher","DOI":"10.2178\/jsl\/1052669062"},{"key":"S021906131750009XBIB007","unstructured":"T. Jech ,  Set Theory. The Third Millennium Edition, Revised and Expanded,  Springer Monographs in Mathematics  (Springer-Verlag,  Berlin,  2003),  xiv+769 pp."},{"key":"S021906131750009XBIB008","first-page":"1","volume":"4","author":"Kurepa D. R.","year":"1935","journal-title":"Publ. Math. Univ. Belgrade"},{"key":"S021906131750009XBIB009","doi-asserted-by":"publisher","DOI":"10.1007\/BF02761175"},{"key":"S021906131750009XBIB010","doi-asserted-by":"publisher","DOI":"10.1016\/j.apal.2007.07.002"},{"key":"S021906131750009XBIB011","doi-asserted-by":"publisher","DOI":"10.1016\/j.apal.2006.05.005"},{"key":"S021906131750009XBIB012","doi-asserted-by":"publisher","DOI":"10.2178\/jsl\/1203350787"},{"key":"S021906131750009XBIB013","doi-asserted-by":"publisher","DOI":"10.1007\/s00153-017-0585-0"},{"key":"S021906131750009XBIB014","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511756306.012"},{"key":"S021906131750009XBIB015","first-page":"130","volume":"1985","author":"Usuba T.","year":"2014","journal-title":"RIMS Kokyuroku"},{"key":"S021906131750009XBIB016","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511910616.003"}],"container-title":["Journal of Mathematical Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S021906131750009X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,6]],"date-time":"2019-08-06T20:34:08Z","timestamp":1565123648000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S021906131750009X"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,11,27]]},"references-count":16,"journal-issue":{"issue":"02","published-online":{"date-parts":[[2017,11,27]]},"published-print":{"date-parts":[[2017,12]]}},"alternative-id":["10.1142\/S021906131750009X"],"URL":"https:\/\/doi.org\/10.1142\/s021906131750009x","relation":{},"ISSN":["0219-0613","1793-6691"],"issn-type":[{"value":"0219-0613","type":"print"},{"value":"1793-6691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,11,27]]}}}