{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,2]],"date-time":"2022-04-02T06:38:39Z","timestamp":1648881519033},"reference-count":5,"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":13798,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1976,6]]},"abstract":"Let \u03b1 be an admissible ordinal, and let (\u03b1) denote the lattice of \u03b1-r.e. sets, ordered by set inclusion. An \u03b1-r.e. set A<\/jats:italic> is \u03b1*-finite<\/jats:italic> if it is \u03b1-finite and has ordertype less than \u03b1* (the \u03a31<\/jats:sub> projectum of \u03b1). An a-r.e. set S<\/jats:italic> is simple<\/jats:italic> if (the complement of S<\/jats:italic>) is not \u03b1*-finite, but all the \u03b1-r.e. subsets of are \u03b1*-finite. Fixing a first-order language \u2112 suitable for lattice theory (see [2, \u00a71] for such a language), and noting that the \u03b1*-finite sets are exactly those elements of (\u03b1), all of whose \u03b1-r.e. subsets have complements in (\u03b1) (see [4, p. 356]), we see that the class of simple \u03b1-r.e. sets is definable in \u2112 over (\u03b1). In [4, \u00a76, (Q22)], we asked whether an admissible ordinal \u03b1 exists for which all simple \u03b1-r.e. sets have the same 1-type. We were particularly interested in this question for \u03b1 = \u21351<\/jats:sub>L<\/jats:sup> (L<\/jats:italic> is G\u00f6del's universe of constructible sets). We will show that for all \u03b1 which are regular cardinals of L<\/jats:italic> (\u21351<\/jats:sub>L<\/jats:sup> is, of course, such an \u03b1), there are simple \u03b1-r.e. sets with different 1-types.<\/jats:p>The sentence exhibited which differentiates between simple \u03b1-r.e. sets is not the first one which comes to mind. Using \u03b1 = \u03c9 for intuition, one would expect any of the sentences \u201cS<\/jats:italic> is a maximal \u03b1-r.e. set\u201d, \u201cS<\/jats:italic> is an r<\/jats:italic>-maximal \u03b1-r.e. set\u201d, or \u201cS<\/jats:italic> is a hyperhypersimple \u03b1-r.e. set\u201d to differentiate between simple \u03b1-r.e. sets. However, if \u03b1 > \u03c9 is a regular cardinal of L<\/jats:italic>, there are no maximal, r<\/jats:italic>-maximal, or hyperhypersimple \u03b1-r.e. sets (see [4, Theorem 4.11], [5, Theorem 5.1] and [1,Theorem 5.21] respectively). But another theorem of (\u03c9) points the way.<\/jats:p>","DOI":"10.1017\/s0022481200051471","type":"journal-article","created":{"date-parts":[[2014,3,13]],"date-time":"2014-03-13T08:40:02Z","timestamp":1394700002000},"page":"419-426","source":"Crossref","is-referenced-by-count":0,"title":["Types of simple \u03b1-recursively enumerable sets"],"prefix":"10.1017","volume":"41","author":[{"given":"Manuel","family":"Lerman","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200051471_ref002","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1968-0227009-1"},{"key":"S0022481200051471_ref001","volume-title":"Annals of Mathematical Logic","author":"Chono"},{"key":"S0022481200051471_ref004","first-page":"341","article-title":"Maximal \u03b1-r.e. sets","volume":"188","author":"Lerman","year":"1974","journal-title":"Transactions of the American Mathematical Society"},{"key":"S0022481200051471_ref005","doi-asserted-by":"publisher","DOI":"10.1007\/BF02764882"},{"key":"S0022481200051471_ref003","first-page":"405","volume":"41","author":"Lerman","year":"1976","journal-title":"Congruence relations, filters, ideals, and definability in lattices of \u03b1-recursively enumerable sets"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200051471","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,28]],"date-time":"2019-05-28T16:49:42Z","timestamp":1559062182000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200051471\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1976,6]]},"references-count":5,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1976,6]]}},"alternative-id":["S0022481200051471"],"URL":"http:\/\/dx.doi.org\/10.1017\/s0022481200051471","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":["Logic","Philosophy"],"published":{"date-parts":[[1976,6]]}}}