{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,10]],"date-time":"2026-03-10T11:33:53Z","timestamp":1773142433620,"version":"3.50.1"},"reference-count":29,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":1197,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2010,12]]},"abstract":"<jats:p><jats:bold>\u00a71. Introduction<\/jats:bold>. It is a well-known phenomenon in set theory that problems in infinite combinatorics involving singular cardinals and their successors tend to be harder than the parallel problems for regular cardinals. Examples include the behaviour of cardinal exponentiation, the extent of the tree property, the extent of stationary reflection, and the existence of non-free almost-free abelian groups. The explanation for this phenomenon lies in inner model theory, in particular <jats:italic>core models<\/jats:italic> and <jats:italic>covering lemmas<\/jats:italic>. If <jats:italic>W<\/jats:italic> is an inner model of <jats:italic>V<\/jats:italic> then<\/jats:p><jats:p>1. <jats:italic>W strongly covers V<\/jats:italic> if every uncountable set of ordinals is covered by a set of the same <jats:italic>V<\/jats:italic> -cardinality lying in <jats:italic>W<\/jats:italic>.<\/jats:p><jats:p>2. <jats:italic>W weakly covers V<\/jats:italic> if <jats:italic>W<\/jats:italic> computes the successor of every <jats:italic>V<\/jats:italic>-singular cardinal correctly.<\/jats:p><jats:p>Strong covering implies weak covering.<\/jats:p><jats:p>In inner model theory there are many theorems of the general form \u201cif there is no inner model of large cardinal hypothesis <jats:italic>X<\/jats:italic> then there is an <jats:italic>L<\/jats:italic>-like inner model K<jats:sub>x<\/jats:sub> which <jats:italic>Y<\/jats:italic> covers <jats:italic>V<\/jats:italic>\u201d. Here the <jats:italic>L<\/jats:italic>-like properties of K<jats:sub>x<\/jats:sub> always include GCH and Global Square. Examples include<\/jats:p><jats:p>1. <jats:italic>X<\/jats:italic> is \u201c0<jats:sup>#<\/jats:sup> exists\u201d, <jats:italic>K<jats:sub>x<\/jats:sub><\/jats:italic> is <jats:italic>L, Y<\/jats:italic> is \u201cstrongly\u201d.<\/jats:p><jats:p>2. <jats:italic>X<\/jats:italic> is \u201cthere is a measurable cardinal\u201d, <jats:italic>K<jats:sub>x<\/jats:sub><\/jats:italic> is the Dodd-Jensen core model, <jats:italic>Y<\/jats:italic> is \u201cstrongly\u201d.<\/jats:p><jats:p>3. <jats:italic>X<\/jats:italic> is \u201cthere is a Woodin cardinal\u201d, <jats:italic>K<jats:sup>x<\/jats:sup><\/jats:italic> is the core model for a Woodin cardinal, <jats:italic>Y<\/jats:italic> is \u201cweakly\u201d.<\/jats:p>","DOI":"10.2178\/jsl\/1286198153","type":"journal-article","created":{"date-parts":[[2010,10,4]],"date-time":"2010-10-04T13:16:13Z","timestamp":1286198173000},"page":"1383-1402","source":"Crossref","is-referenced-by-count":14,"title":["Diagonal Prikry extensions"],"prefix":"10.1017","volume":"75","author":[{"given":"James","family":"Cummings","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Matthew","family":"Foreman","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200002309_ref029","unstructured":"Woodin W. 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