{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,3]],"date-time":"2022-04-03T10:35:09Z","timestamp":1648982109784},"reference-count":11,"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":12611,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1979,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Let <jats:italic>A<\/jats:italic> be a standard transitive admissible set. \u03a3<jats:sub>1<\/jats:sub>-separation is the principle that whenever <jats:italic>X<\/jats:italic> and <jats:italic>Y<\/jats:italic> are disjoint \u03a3<jats:sub arrange=\"stack\">1<\/jats:sub><jats:sup arrange=\"stack\">A<\/jats:sup> subsets of <jats:italic>A<\/jats:italic> then there is a \u22bf<jats:sub arrange=\"stack\">1<\/jats:sub><jats:sup arrange=\"stack\">A<\/jats:sup> subset <jats:italic>S<\/jats:italic> of <jats:italic>A<\/jats:italic> such that <jats:italic>X<\/jats:italic> \u2286 <jats:italic>S<\/jats:italic> and <jats:italic>Y<\/jats:italic> \u2229 <jats:italic>S<\/jats:italic> = \u2205.<\/jats:p><jats:p>Theorem. <jats:italic>If satisfies \u03a3-separation, then<\/jats:italic><\/jats:p><jats:p>(1) <jats:italic>If \u3008T<jats:sub>n<\/jats:sub>\u2223n &lt; \u03c9) \u03f5 A is a sequence of trees on \u03c9 each of which has at most finitely many infinite paths in A then the function n \u21a6 (set of infinite paths in A through T<jats:sub>n<\/jats:sub>) is in A<\/jats:italic>.<\/jats:p><jats:p>(2) <jats:italic>If A is not closed under hyperjump and \u03b1 = On<jats:sup>A<\/jats:sup>  then A has in it a nonstandard model of V = L whose ordinal standard part is \u03b1<\/jats:italic>.<\/jats:p><jats:p>Theorem. Let \u03b1 be any countable admissible ordinal greater than \u03c9. Then there is a model of \u03a3<jats:sub>1<\/jats:sub>-separation whose height is \u03b1.<\/jats:p>","DOI":"10.2307\/2273130","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:50:56Z","timestamp":1146937856000},"page":"374-382","source":"Crossref","is-referenced-by-count":0,"title":["\u03a3<sub>1<\/sub>-separation"],"prefix":"10.1017","volume":"44","author":[{"given":"Fred G.","family":"Abramson","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200048295_ref010","unstructured":"Simpson S. G. , Notes on subsystems of analysis, mimeographed, 1973."},{"key":"S0022481200048295_ref008","volume-title":"Theory of recursive functions and effective computability","author":"Rogers","year":"1967"},{"key":"S0022481200048295_ref004","first-page":"113","volume":"40","author":"Friedman","year":"1975","journal-title":"One hundred and two problems in mathematical logic"},{"key":"S0022481200048295_ref002","unstructured":"Abramson F. G. , Locally countable models of \u03a31-separation (to appear)."},{"key":"S0022481200048295_ref001","volume-title":"Advances in Mathematics","author":"Abramson"},{"key":"S0022481200048295_ref009","first-page":"132","volume-title":"Essays on the foundations of mathematics","author":"Shoenfield","year":"1961"},{"key":"S0022481200048295_ref005","unstructured":"Harrington L. A. , An admissible set with no intermediate \u03a31-degrees (in preparation)."},{"key":"S0022481200048295_ref011","first-page":"226","volume":"34","author":"Barwise","year":"1969","journal-title":"Infinitary logic and admissible sets"},{"key":"S0022481200048295_ref006","first-page":"84","volume-title":"Mathematical logic and foundation of set theory, Proceedings of an International Colloquium, Jerusalem","author":"Jensen","year":"1968"},{"key":"S0022481200048295_ref007","volume-title":"Model theory for infinitary logic","author":"Keisler","year":"1971"},{"key":"S0022481200048295_ref003","first-page":"539","volume-title":"Cambridge Summer School in Mathematical Logic, 1971, Lecture Notes in Mathematics","author":"Friedman","year":"1973"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200048295","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,26]],"date-time":"2019-05-26T16:52:40Z","timestamp":1558889560000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200048295\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1979,9]]},"references-count":11,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1979,9]]}},"alternative-id":["S0022481200048295"],"URL":"https:\/\/doi.org\/10.2307\/2273130","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1979,9]]}}}