{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,30]],"date-time":"2022-03-30T18:06:44Z","timestamp":1648663604752},"reference-count":4,"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":12520,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1979,12]]},"abstract":"<jats:p>It is well known that the first strongly inaccessible cardinal is strictly less than the first weakly compact cardinal which in turn is strictly less than the first Ramsey cardinal, etc. However, once one passes the first measurable cardinal the inequalities are no longer strict. Magidor [3] has shown that the first strongly compact cardinal may be equal to the first measurable cardinal or equal to the first super-compact cardinal (the first supercompact cardinal is strictly larger than the first measurable cardinal). In this note we will indicate how Magidor's methods can be used to show that it is undecidable whether one cardinal (the first strongly compact) is greater than or less than another large cardinal (the first huge cardinal). We assume that the reader is familiar with the ultrapower construction of Scott, as presented in Drake [1] or Kanamori, Reinhardt and Solovay [2].<\/jats:p><jats:p>Definition. <jats:italic>A<\/jats:italic> cardinal \u03ba is <jats:italic>huge<\/jats:italic> (or 1-huge) if there is an elementary embedding <jats:italic>j<\/jats:italic> of the universe <jats:italic>V<\/jats:italic> into a transitive class <jats:italic>M<\/jats:italic> such that <jats:italic>M<\/jats:italic> contains the ordinals, is closed under <jats:italic>j<\/jats:italic>(\u03ba) sequences, <jats:italic>j<\/jats:italic>(\u03ba) &gt; \u03ba and <jats:italic>j<\/jats:italic> \u21be <jats:italic>R<\/jats:italic><jats:sub>\u03ba<\/jats:sub> = id. Let \u03ba denote the first huge cardinal, and let \u03bb = <jats:italic>j<\/jats:italic>(\u03ba).<\/jats:p><jats:p>One can see from easy reflection arguments that \u03ba and \u03bb are inaccessible in <jats:italic>V<\/jats:italic> and, in fact, that \u03ba is measurable in <jats:italic>V<\/jats:italic>.<\/jats:p>","DOI":"10.2307\/2273295","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:51:21Z","timestamp":1146937881000},"page":"563-565","source":"Crossref","is-referenced-by-count":0,"title":["On the ordering of certain large cardinals"],"prefix":"10.1017","volume":"44","author":[{"given":"Carl F.","family":"Morgenstern","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200048441_ref001","volume-title":"Set theory: an introduction to large cardinals","author":"Drake","year":"1974"},{"key":"S0022481200048441_ref002","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1016\/0003-4843(78)90031-1","article-title":"Strong axioms of infinity and elementary embeddings","volume":"13","author":"Kanamori","year":"1978","journal-title":"Annals of Mathematical Logic"},{"key":"S0022481200048441_ref003","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(76)90024-3"},{"key":"S0022481200048441_ref004","unstructured":"Menas T. , On strong compactness and supercompactness, Ph.D. dissertation, University of California, Berkeley, 1973."}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200048441","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,26]],"date-time":"2019-05-26T16:27:58Z","timestamp":1558888078000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200048441\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1979,12]]},"references-count":4,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1979,12]]}},"alternative-id":["S0022481200048441"],"URL":"https:\/\/doi.org\/10.2307\/2273295","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1979,12]]}}}