{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,2]],"date-time":"2026-03-02T22:50:35Z","timestamp":1772491835256,"version":"3.50.1"},"reference-count":4,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":14256,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1975,3]]},"abstract":"<jats:p>Our unexplained notation is that of Rogers [4]. Let <jats:italic>P<\/jats:italic> \u2286 2<jats:sup>N<\/jats:sup> \u00d7 2<jats:sup>N<\/jats:sup>. We call a sequence &lt;<jats:italic>A<jats:sub>n<\/jats:sub>: n \u2208 N<\/jats:italic>&gt; of subsets of <jats:italic>N<\/jats:italic> a <jats:italic>P-sequence<\/jats:italic> iff <jats:italic>\u2200n(A<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic>+1<\/jats:sub> = the unique <jats:italic>B<\/jats:italic> such that <jats:italic>P(A<jats:sub>n<\/jats:sub>, B))<\/jats:italic>.<\/jats:p><jats:p>Theorem. <jats:italic>Let P<\/jats:italic> \u2286 2<jats:italic><jats:sup>N<\/jats:sup><\/jats:italic> \u00d7 2<jats:italic><jats:sup>N<\/jats:sup> be arithmetical. Then there is no P-sequence &lt;A<jats:sub>n<\/jats:sub>: n \u2208 N&gt; such that \u2200n(A\u2032<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic>+1<\/jats:sub> \u2264<jats:italic><jats:sub>T<\/jats:sub> A<jats:sub>n<\/jats:sub>)<\/jats:italic>.<\/jats:p><jats:p>This theorem improves a result of Friedman [2] who showed that for no arithmetical <jats:italic>P<\/jats:italic> is there a <jats:italic>P<\/jats:italic>-sequence &lt;<jats:italic>A<jats:sub>n<\/jats:sub>: n \u2208 N<\/jats:italic>&gt; such that <jats:italic>A<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic> + 1<\/jats:sub> is a code for an \u03c9-model of the relative arithmetic comprehension schema, and <jats:italic>A<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic> + 1<\/jats:sub> is present in the model coded by <jats:italic>A<jats:sub>n<\/jats:sub><\/jats:italic>, for all <jats:italic>n<\/jats:italic>. Other related results are those of Harrison [3], who showed there is a sequence &lt;<jats:italic>A<jats:sub>n<\/jats:sub>: n \u2208 N<\/jats:italic>&gt; such that <jats:italic>\u2200n&lt;A\u2032<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic> + 1<\/jats:sub> \u2264<jats:italic><jats:sub>T<\/jats:sub> A<jats:sub>n<\/jats:sub><\/jats:italic>&gt;, and of Enderton and Putnam [1], who showed there is no sequence &lt;<jats:italic>A<jats:sub>n<\/jats:sub>: n \u2208 N<\/jats:italic>&gt; with <jats:italic>\u2200n(A\u2032<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic> + 1<\/jats:sub> \u2264<jats:italic><jats:sub>T<\/jats:sub> A<jats:sub>n<\/jats:sub>)<\/jats:italic> and <jats:italic>A<jats:sub>0<\/jats:sub><\/jats:italic> hyperarithmetic.<\/jats:p><jats:p>Our theorem is closely connected to G\u00f6del's second incompleteness theorem. Its proof is a recursion theoretic parallel to the proof of G\u00f6del's theorem. In \u00a72 we draw a version of G\u00f6del's theorem as a corollary to ours.<\/jats:p>","DOI":"10.2307\/2272271","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:33:55Z","timestamp":1146951235000},"page":"59-61","source":"Crossref","is-referenced-by-count":8,"title":["Descending sequences of degrees"],"prefix":"10.1017","volume":"40","author":[{"given":"John","family":"Steel","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S002248120005427X_ref003","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1968-0244049-7"},{"key":"S002248120005427X_ref002","unstructured":"Friedman H. , Sequences of models (unpublished)."},{"key":"S002248120005427X_ref001","first-page":"429","volume":"35","author":"Enderton","year":"1970","journal-title":"A note on the hyperarithmetical hierarchy"},{"key":"S002248120005427X_ref004","volume-title":"Theory of recursive functions and effective computability","author":"Rogers","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\/S002248120005427X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,29]],"date-time":"2019-05-29T20:06:06Z","timestamp":1559160366000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S002248120005427X\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1975,3]]},"references-count":4,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1975,3]]}},"alternative-id":["S002248120005427X"],"URL":"https:\/\/doi.org\/10.2307\/2272271","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1975,3]]}}}