{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,6]],"date-time":"2026-04-06T22:33:29Z","timestamp":1775514809646,"version":"3.50.1"},"reference-count":8,"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":12976,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1978,9]]},"abstract":"<jats:p>In recent years, the Axiom of Determinateness (AD) has yielded numerous results concerning the size and properties of the first \u03c9-many uncountable cardinals. Briefly, these results began with Solovay's discovery that \u2135<jats:sub>1<\/jats:sub> and \u2135<jats:sub>2<\/jats:sub> are measurable [8], [3], continued with theorems of Solovay, Martin, and Kunen concerning infinite-exponent partition relations [6], [3], Martin's proof that \u2135<jats:sub><jats:italic>n<\/jats:italic><\/jats:sub> has confinality \u2135<jats:sub>2<\/jats:sub> for 1 &lt; <jats:italic>n<\/jats:italic> &lt; \u03c9, and very recently, Kleinberg's proof that the \u2135<jats:sub><jats:italic>n<\/jats:italic><\/jats:sub> are Jonsson cardinals [4].<\/jats:p><jats:p>This paper was inspired by a very recent result of Martin from AD that \u2135<jats:sub>1<\/jats:sub> is \u2135<jats:sub>2<\/jats:sub>-super compact. It was known for some time that AD implies \u2135<jats:sub>1<\/jats:sub> is \u03b1-strongly compact for all \u2135 &lt; \u03b8 (where \u03b8 is the least cardinal onto which 2<jats:sup>\u03c9<\/jats:sup> cannot be mapped, quite a large cardinal under AD), and that AD<jats:sub>R<\/jats:sub> implies that \u2135<jats:sub>1<\/jats:sub>, is \u03b1-super compact for all \u03b1 &lt; \u03b8. A key open question had been whether or not \u2135<jats:sub>1<\/jats:sub> is super compact under AD alone.<\/jats:p><jats:p>This paper comments on the method of Martin in several different ways. In \u00a72, we will prove that \u2135<jats:sub>1<\/jats:sub> is \u2135<jats:sub>2<\/jats:sub>-super compact, and then generalize the method to show that \u2135<jats:sub>2<\/jats:sub> is \u2135<jats:sub>3<\/jats:sub>-strongly compact. In addition, we will demonstrate a limitation in the method by showing that the possible measures obtained on <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120004929X_inline1\"\/> are not normal, and that the method cannot be extended to show that \u2135<jats:sub>2<\/jats:sub> is \u2135<jats:sub>4<\/jats:sub>-strongly compact.<\/jats:p>","DOI":"10.2307\/2273517","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:48:21Z","timestamp":1146952101000},"page":"394-401","source":"Crossref","is-referenced-by-count":9,"title":["On the compactness of \u2135<sub>1<\/sub> and \u2135<sub>2<\/sub>"],"prefix":"10.1017","volume":"43","author":[{"given":"C. A.","family":"di Prisco","sequence":"first","affiliation":[]},{"given":"J.","family":"Henle","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S002248120004929X_ref005","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1974-0327518-9"},{"key":"S002248120004929X_ref001","unstructured":"Di Prisco C. , Combinatorial properties and supercompact cardinals, Doctoral Dissertation, M.I.T., 1976."},{"key":"S002248120004929X_ref002","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(73)90014-4"},{"key":"S002248120004929X_ref007","unstructured":"Menas T. K. , On strong compactness and super compactness, Doctoral Dissertation, University of California, Berkeley."},{"key":"S002248120004929X_ref003","unstructured":"Kechris A. , Notes prepared for the M.I.T. Logic Seminar, 1972\u20131973."},{"key":"S002248120004929X_ref006","unstructured":"Martin D. A. , Determinateness implies many cardinals are measurable, mimeographed."},{"key":"S002248120004929X_ref004","volume-title":"Annals of Mathematical Logic","author":"Kleinberg"},{"key":"S002248120004929X_ref008","volume-title":"Measurable cardinals and the axiom of determinateness","author":"Solovay","year":"1967"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S002248120004929X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,27]],"date-time":"2019-05-27T19:33:41Z","timestamp":1558985621000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S002248120004929X\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1978,9]]},"references-count":8,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1978,9]]}},"alternative-id":["S002248120004929X"],"URL":"https:\/\/doi.org\/10.2307\/2273517","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1978,9]]}}}