{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,6]],"date-time":"2026-04-06T20:16:29Z","timestamp":1775506589780,"version":"3.50.1"},"reference-count":19,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":10876,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1984,6]]},"abstract":"<jats:p>Let PA be the theory of first order Peano arithmetic, in the language <jats:italic>L<\/jats:italic> with binary operation symbols + and \u00b7. Let <jats:italic>N<\/jats:italic> be the theory of the standard model of PA. We consider countable models <jats:italic>M<\/jats:italic> of PA such that the universe \u2223<jats:italic>M<\/jats:italic>\u2223 is <jats:italic>\u03c9<\/jats:italic>. The <jats:italic>degree<\/jats:italic> of such a model <jats:italic>M<\/jats:italic>, denoted by deg(<jats:italic>M<\/jats:italic>), is the (Turing) degree of the atomic diagram of <jats:italic>M<\/jats:italic>. The results of this paper concern the degrees of models of <jats:italic>N<\/jats:italic>, but here in the Introduction, we shall give a brief survey of results about degrees of models of PA.<\/jats:p><jats:p>Let <jats:bold>D<\/jats:bold><jats:sub>0<\/jats:sub> denote the set of degrees <jats:bold>d<\/jats:bold> such that there is a nonstandard model of <jats:italic>M<\/jats:italic> of PA with deg(<jats:italic>M<\/jats:italic>) = <jats:bold>d<\/jats:bold>. Here are some of the more easily stated results about <jats:bold>D<\/jats:bold><jats:sub>0<\/jats:sub>.<\/jats:p><jats:p>(1) <jats:italic>There is no recursive nonstandard model of PA; i.e.<\/jats:italic>, <jats:bold>0<\/jats:bold> \u2208 <jats:bold>D<\/jats:bold><jats:sub>0<\/jats:sub>.<\/jats:p><jats:p>This is a result of Tennenbaum [T].<\/jats:p><jats:p>(2) <jats:italic>There exists<\/jats:italic><jats:bold>d<\/jats:bold> \u2208 <jats:bold>D<\/jats:bold><jats:sub>0<\/jats:sub><jats:italic>such that<\/jats:italic><jats:bold>d<\/jats:bold> \u2264 <jats:bold>0<\/jats:bold>\u2032.<\/jats:p><jats:p>This follows from the standard Henkin argument.<\/jats:p><jats:p>(3) <jats:italic>There exists<\/jats:italic><jats:bold>d<\/jats:bold> \u2208 <jats:bold>D<\/jats:bold><jats:sub>0<\/jats:sub><jats:italic>such that<\/jats:italic><jats:bold>d<\/jats:bold> &lt; <jats:bold>0<\/jats:bold>\u2032.<\/jats:p><jats:p>Shoenfield [Sh1] proved this, using the Kreisel-Shoenfield basis theorem.<\/jats:p><jats:p>(4) <jats:italic>There exists<\/jats:italic><jats:bold>d<\/jats:bold> \u2208 <jats:bold>D<\/jats:bold><jats:sub>0<\/jats:sub><jats:italic>such that<\/jats:italic><jats:bold>d<\/jats:bold>\u2032 = <jats:bold>0<\/jats:bold>\u2032.<\/jats:p><jats:p>Jockusch and Soare [JS] improved the Kreisel-Shoenfield basis theorem and obtained (4).<\/jats:p><jats:p>(5) <jats:bold>D<\/jats:bold><jats:sub>0<\/jats:sub> = <jats:bold>D<\/jats:bold><jats:sub><jats:italic>c<\/jats:italic><\/jats:sub> = <jats:bold>D<\/jats:bold><jats:sub><jats:italic>e<\/jats:italic><\/jats:sub>, where <jats:bold>D<\/jats:bold><jats:sub><jats:italic>c<\/jats:italic><\/jats:sub> denotes the set of degrees of completions of PA and <jats:bold>D<\/jats:bold><jats:sub><jats:italic>e<\/jats:italic><\/jats:sub> the set of degrees <jats:bold>d<\/jats:bold> such that <jats:bold>d<\/jats:bold> separates a pair of effectively inseparable r.e. sets.<\/jats:p><jats:p>Solovay noted (5) in a letter to Soare in which in answer to a question posed in [JS] he showed that <jats:bold>D<\/jats:bold><jats:sub><jats:italic>c<\/jats:italic><\/jats:sub> is upward closed.<\/jats:p>","DOI":"10.2307\/2274174","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:08:28Z","timestamp":1146953308000},"page":"425-436","source":"Crossref","is-referenced-by-count":8,"title":["Two theorems on degrees of models of true arithmetic"],"prefix":"10.1017","volume":"49","author":[{"given":"Julia","family":"Knight","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alistair H.","family":"Lachlan","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Robert I.","family":"Soare","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200033478_ref019","first-page":"270","article-title":"Non-archimedean models for arithmetic","volume":"6","author":"Tennenbaum","year":"1959","journal-title":"Notices of the American Mathematical Society"},{"key":"S0022481200033478_ref011","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19660120125"},{"key":"S0022481200033478_ref015","doi-asserted-by":"publisher","DOI":"10.1090\/pspum\/005\/0141595"},{"key":"S0022481200033478_ref004","volume-title":"Building nonstandard models of Peano arithmetic","author":"Harrington","year":"1979"},{"key":"S0022481200033478_ref001","first-page":"429","volume":"35","author":"Enderton","year":"1970","journal-title":"A note on the hyperarithmetic 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