{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,6,13]],"date-time":"2022-06-13T12:30:27Z","timestamp":1655123427066},"reference-count":3,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":7497,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1993,9]]},"abstract":"Let E<\/jats:italic> \u22b2 F<\/jats:italic> iff E<\/jats:italic> and F<\/jats:italic> are extenders and E<\/jats:italic> \u2208 Ult(V, F<\/jats:italic>). Intuitively, E<\/jats:italic> \u22b2 F<\/jats:italic> implies that E<\/jats:italic> is weaker\u2014embodies less reflection\u2014than F<\/jats:italic>. The relation \u22b2 was first considered by W. Mitchell in [M74], where it arises naturally in connection with inner models and coherent sequences. Mitchell showed in [M74] that the restriction of \u22b2 to normal ultrafilters is well-founded.<\/jats:p>The relation \u22b2 is now known as the Mitchell order, although it is not actually an order. It is irreflexive, and its restriction to normal ultrafilters is transitive, but under mild large cardinal hypotheses, it is not transitive on all extenders. Here is a counterexample. Let \u03ba be (\u03bb + 2)-strong, where \u03bb > \u03ba and \u03bb is measurable. Let E<\/jats:italic> be an extender with critical point \u03ba<\/jats:italic> and let U<\/jats:italic> be a normal ultrafilter with critical point \u03bb such that U<\/jats:italic> \u2208 Ult(V, E<\/jats:italic>). Let i: V<\/jats:italic> \u2192 Ult(V, U<\/jats:italic>) be the canonical embedding. Then i<\/jats:italic>(E<\/jats:italic>) \u22b2 U<\/jats:italic> and U<\/jats:italic> \u22b2 E<\/jats:italic>, but by 3.11 of [MS2], it is not the case that i<\/jats:italic>(E<\/jats:italic>) \u22b2 E<\/jats:italic>. (The referee pointed out the following elementary proof of this fact. Notice that i<\/jats:italic> \u21be V\u03bb+2<\/jats:sub> \u2208 Ult(V, E<\/jats:italic>) and X<\/jats:italic> \u2208 E<\/jats:italic>a<\/jats:italic><\/jats:sub> \u21d4 X<\/jats:italic> \u2208 i<\/jats:italic>(E<\/jats:italic>)i<\/jats:italic>(a<\/jats:italic>)<\/jats:sub>. Moreover, we may assume without loss of generality that = support(E<\/jats:italic>). Thus, if i<\/jats:italic>(E<\/jats:italic>) \u2208 Ult(V, E<\/jats:italic>), then E<\/jats:italic> \u2208 Ult(V, E<\/jats:italic>), a contradiction.)<\/jats:p>By going to much stronger extenders, one can show the Mitchell order is not well-founded. The following example is well known. Let j<\/jats:italic>: V<\/jats:italic> \u2192 M<\/jats:italic> be elementary, with V<\/jats:italic>\u03bb<\/jats:sub> \u2286 M<\/jats:italic> for \u03bb = j<\/jats:italic>o\u03c9<\/jats:sub>(crit(j)). (By Kunen, V<\/jats:italic>\u03bb+1<\/jats:sub> \u2209 M<\/jats:italic>.) Let E<\/jats:italic>0<\/jats:sub> be the (crit(j), \u03bb) extender derived from j<\/jats:italic>, and let E<\/jats:italic>n<\/jats:italic>+1<\/jats:sub> = i<\/jats:italic>(En<\/jats:sub><\/jats:italic>), where i<\/jats:italic>: V<\/jats:italic> \u2192 Ult(V<\/jats:italic>, En<\/jats:sub><\/jats:italic>) is the canonical embedding. One can show inductively that En<\/jats:sub><\/jats:italic> is an extender over V<\/jats:italic>, and thereby, that E<\/jats:italic>n+1<\/jats:sub> \u22b2 En<\/jats:sub><\/jats:italic> for all n<\/jats:italic> < \u03c9. (There is a little work in showing that Ult(V<\/jats:italic>, E<\/jats:italic>n<\/jats:italic>+1<\/jats:sub>) is well-founded.)<\/jats:p>","DOI":"10.2307\/2275105","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:49:16Z","timestamp":1146955756000},"page":"931-940","source":"Crossref","is-referenced-by-count":12,"title":["The well-foundedness of the Mitchell order"],"prefix":"10.1017","volume":"58","author":[{"given":"J. R.","family":"Steel","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200020971_ref002","doi-asserted-by":"publisher","DOI":"10.1090\/S0894-0347-1989-0955605-X"},{"key":"S0022481200020971_ref001","first-page":"57","author":"Mitchell","journal-title":"Sets constructive from sequences of ultrafilters"},{"key":"S0022481200020971_ref003","article-title":"Iteration trees","author":"Martin","journal-title":"Journal of the American Mathematical Society"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200020971","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,15]],"date-time":"2019-05-15T22:02:55Z","timestamp":1557957775000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200020971\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1993,9]]},"references-count":3,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1993,9]]}},"alternative-id":["S0022481200020971"],"URL":"http:\/\/dx.doi.org\/10.2307\/2275105","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":["Logic","Philosophy"],"published":{"date-parts":[[1993,9]]}}}