{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,1]],"date-time":"2022-04-01T21:03:11Z","timestamp":1648846991312},"reference-count":8,"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":9781,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1987,6]]},"abstract":"We shall be concerned here with weak axiomatic systems of set theory with a universal set. The language in which they are expressed is that of set theory\u2014two primitive predicates, = and \u03f5, and no function symbols (though some function symbols will be introduced by definitional abbreviation). All the theories will have stratified axioms only, and they will all have Ext (extensionality: (\u2200x<\/jats:italic>)(\u2200y<\/jats:italic>)(x = y<\/jats:italic>\u00b7 \u2194 \u00b7(\u2200z<\/jats:italic>)(z<\/jats:italic> \u03f5 x<\/jats:italic> \u2194 z<\/jats:italic> \u03f5 y<\/jats:italic>))). In fact, in addition to extensionality, they have only axioms saying that the universe is closed under certain set-theoretic operations, viz. all of the form<\/jats:p><\/jats:disp-formula><\/jats:p>and these will always include singleton, i.e., \u03b9<\/jats:italic>\u2032x<\/jats:italic> exists if x<\/jats:italic> does (the iota notation for singleton, due to Russell and Whitehead, is used here to avoid confusion with {x<\/jats:italic>: \u03a6<\/jats:italic>}, set abstraction), and also x<\/jats:italic> \u222a y<\/jats:italic>, x<\/jats:italic> \u2229 y<\/jats:italic> and \u2212 x<\/jats:italic> (the complement of x<\/jats:italic>). The system with these axioms is called NF2<\/jats:sub> in the literature (see [F]). The other axioms we consider will be those giving \u22c3x<\/jats:italic>, \u22c2x<\/jats:italic>, {y<\/jats:italic>: y<\/jats:italic> \u2286x<\/jats:italic>} and {y<\/jats:italic>: x<\/jats:italic> \u2286 y<\/jats:italic>}. We will frequently have occasion to bear in mind that \u3008 V<\/jats:italic>, \u2286 \u3009 is a Boolean algebra in any theory extending NF2<\/jats:sub>. There is no use of the axiom of choice at any point in this paper. Since the systems with which we will be concerned exhibit this feature of having, in addition to extensionality, only axioms stating that V<\/jats:italic> is closed under certain operations, we will be very interested in terms<\/jats:italic> of the theories in question. A T<\/jats:italic>-term, for T<\/jats:italic> such a theory, is a thing (with no free variables) built up from V<\/jats:italic> or \u2227<\/jats:italic> by means of the T<\/jats:italic>-operations, which are of course the operations that the axioms of T<\/jats:italic> say the universe is closed under.<\/jats:p>","DOI":"10.2307\/2274387","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:21:40Z","timestamp":1146954100000},"page":"374-387","source":"Crossref","is-referenced-by-count":3,"title":["Term models for weak set theories with a universal set"],"prefix":"10.1017","volume":"52","author":[{"given":"T. E.","family":"Forster","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200034332_ref005","doi-asserted-by":"crossref","DOI":"10.1090\/memo\/0057","volume-title":"A hierarchy of formulas for set theory","author":"L\u00e9vy","year":"1965"},{"key":"S0022481200034332_ref008","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1002\/malq.19820281303","article-title":"A decision method for the universal theorems of Quine's New Foundations","volume":"28","author":"Oswald","year":"1982","journal-title":"Zeitschrift f\u00fcr Mathematische Logik und Grundlagen der Mathematik"},{"key":"S0022481200034332_ref006","volume-title":"A model of set theory with a universal set","author":"Mitchell","year":"1976"},{"key":"S0022481200034332_ref004","first-page":"1","volume":"9","author":"Hailperin","year":"1944","journal-title":"A set of axioms for logic"},{"key":"S0022481200034332_ref007","first-page":"23","volume-title":"La th\u00e9orie des ensembles de Quine (Colloque, Louvain-La-Neuve, 1981)","author":"Oswald","year":"1983"},{"key":"S0022481200034332_ref002","first-page":"131","volume":"47","author":"Crabb\u00e9","year":"1982","journal-title":"The consistency of an impredicative subsystem of Quine's NF"},{"key":"S0022481200034332_ref003","volume-title":"Quine's NF, an introduction","author":"Forster","year":"1983"},{"key":"S0022481200034332_ref001","first-page":"1657","article-title":"Les th\u00e9oremes 3-stratifi\u00e9s de NF","volume":"280","author":"Boffa","year":"1975","journal-title":"Comptes Rendus Hebdomadaires des S\u00e9ances de l'Acad\u00e9mic des Sciences (Paris), S\u00e9rie A"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200034332","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,21]],"date-time":"2019-05-21T19:48:22Z","timestamp":1558468102000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200034332\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1987,6]]},"references-count":8,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1987,6]]}},"alternative-id":["S0022481200034332"],"URL":"https:\/\/doi.org\/10.2307\/2274387","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1987,6]]}}}