{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,25]],"date-time":"2025-02-25T13:58:15Z","timestamp":1740491895114,"version":"3.38.0"},"reference-count":7,"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":13890,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1976,3]]},"abstract":"<jats:p>The constructible universe <jats:italic>L<\/jats:italic> of G\u00f6del [2] has a natural well-ordering &lt;<jats:sub><jats:italic>L<\/jats:italic><\/jats:sub> given by the order of construction; a closer look reveals that this well-ordering is definable by a \u03a3<jats:sub>1<\/jats:sub> formula. Cohen's method of forcing provides several examples of models of <jats:italic>ZF<\/jats:italic> + <jats:italic>V<\/jats:italic> \u2260 <jats:italic>L<\/jats:italic> which have a definable well-ordering but none is definable by a relatively simple formula.<\/jats:p><jats:p>Recently, Mansfield [7] has shown that if a set of reals (or hereditarily countable sets) has a \u03a3<jats:sub>1<\/jats:sub>, well-ordering then each of its elements is constructible. A question has thus arisen whether one can find a model of ZF + <jats:italic>V<\/jats:italic> \u2260 <jats:italic>L<\/jats:italic> that has a \u03a3<jats:sub>1<\/jats:sub> well-ordering of the universe. We answer this question in the affirmative.<\/jats:p><jats:p>The main result of this paper is<\/jats:p><jats:p>Theorem. <jats:italic>There is a model of<\/jats:italic> ZF + <jats:italic>V<\/jats:italic> \u2260 <jats:italic>L which has a \u03a3<jats:sub>1<\/jats:sub> well-ordering<\/jats:italic>.<\/jats:p><jats:p>The model is a generic extension of <jats:italic>L<\/jats:italic> by adjoining a branch through a Suslin tree with certain properties. The branch is a nonconstructible subset of \u2135<jats:sub>1<\/jats:sub>. Note that by Mansfield's theorem, the model must not have nonconstructible subsets of \u03c9.<\/jats:p><jats:p>Our results can be generalized in several directions. We note that in particular, we can get a model with a \u03a3<jats:sub>1<\/jats:sub> well-ordering that is not <jats:italic>L[X]<\/jats:italic> for any set <jats:italic>X<\/jats:italic>. As one might expect from a joint paper by a recursion theorist and a set theorist, the proof consists of a construction and a computation.<\/jats:p>","DOI":"10.1017\/s0022481200051860","type":"journal-article","created":{"date-parts":[[2014,3,13]],"date-time":"2014-03-13T08:42:09Z","timestamp":1394700129000},"page":"167-170","source":"Crossref","is-referenced-by-count":0,"title":["On \u03a3<sub>1<\/sub> well-orderings of the universe"],"prefix":"10.1017","volume":"41","author":[{"given":"Leo","family":"Harrington","sequence":"first","affiliation":[]},{"given":"Thomas","family":"Jech","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200051860_ref004","first-page":"122","volume-title":"Mathematical Logic and Set Theory Proceedings","author":"Jensen","year":"1968"},{"key":"S0022481200051860_ref005","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(72)90001-0"},{"key":"S0022481200051860_ref001","first-page":"79","volume":"39","author":"Friedman","year":"1974","journal-title":"PCA well-ordering of the line"},{"key":"S0022481200051860_ref006","first-page":"396","volume":"38","author":"Mansfield","year":"1973","journal-title":"On the possibility of a \u03a321 well-ordering of the Baire space"},{"volume-title":"The consistency of the axiom of choice and of the generalized continuum hypothesis","year":"1940","author":"G\u00f6del","key":"S0022481200051860_ref002"},{"key":"S0022481200051860_ref007","unstructured":"Mansfield R. , The nonexistence of \u03a32 1 well-orderings of the Baire space (to appear)."},{"journal-title":"Israel Journal of Mathematics","article-title":"Simple complete Boolean algebras","author":"Jech","key":"S0022481200051860_ref003"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200051860","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,28]],"date-time":"2019-05-28T17:12:19Z","timestamp":1559063539000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200051860\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1976,3]]},"references-count":7,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1976,3]]}},"alternative-id":["S0022481200051860"],"URL":"https:\/\/doi.org\/10.1017\/s0022481200051860","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"type":"print","value":"0022-4812"},{"type":"electronic","value":"1943-5886"}],"subject":[],"published":{"date-parts":[[1976,3]]}}}