{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T20:56:47Z","timestamp":1775854607993,"version":"3.50.1"},"reference-count":13,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2006,11,13]],"date-time":"2006-11-13T00:00:00Z","timestamp":1163376000000},"content-version":"vor","delay-in-days":5430,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematical Logic Qtrly"],"published-print":{"date-parts":[[1992,1]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>The still unsettled decision problem for the restricted purely universal formulae ((\u2200)<jats:sub>0<\/jats:sub>\u2010formulae) of the first order set\u2010theoretic language based over =, \u2208 is discussed in relation with the adoption or rejection of the axiom of foundation. Assuming the axiom of foundation, the related finite set\u2010satisfiability problem for the very significant subclass of the (\u2200)<jats:sub>0<\/jats:sub>\u2010formulae consisting of the formulae involving only nested variables of level 1 is proved to be semidecidable on the ground of a reflection property over the hereditarily finite sets, and various extensions of this result are obtained. When variables are restricted to range only over sets, in universes with infinitely many urelements the set\u2010satisfiability problem is shown to be solvable provided the axiom of foundation is assumed; if it is not, then the decidability of a related derivability problem still holds. That, in turn, suggests the alternative adoption of an antifoundation axiom under which the set\u2010satisfiability problem is also solvable (of course with different answers). Turning to set theory without urelements, assuming a form of Boffa's antifoundation axiom, the complement of the set\u2010satisfiability problem for the full class of \u0394<jats:sub>0<\/jats:sub>\u2010formulae is shown to be semidecidable; a result that is known not to hold, for the set\u2010satisfiability problem itself, even for a very restricted subclass of the \u0394<jats:sub>0<\/jats:sub>\u2010formulae.<\/jats:p>","DOI":"10.1002\/malq.19920380110","type":"journal-article","created":{"date-parts":[[2007,5,29]],"date-time":"2007-05-29T05:45:42Z","timestamp":1180417542000},"page":"143-156","source":"Crossref","is-referenced-by-count":4,"title":["THE DECISION PROBLEM FOR RESTRICTED UNIVERSAL QUANTIFICATION IN SET THEORY AND THE AXIOM OF FOUNDATION"],"prefix":"10.1002","volume":"38","author":[{"given":"Franco","family":"Parlamento","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alberto","family":"Policriti","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2006,11,13]]},"reference":[{"key":"e_1_2_1_2_2","unstructured":"Aczel P. Non\u2010well founded sets. CSLI Lecture Notes 14 Stanford Ca. 1988."},{"key":"e_1_2_1_3_2","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.3160340203"},{"key":"e_1_2_1_4_2","unstructured":"Cantone D. V.Cutello andA.Policriti Decidability results for classes of purely universal formulae and quantifiers elimination in set theory. Le Matematiche Catania1988."},{"key":"e_1_2_1_5_2","unstructured":"Cantone D. V.Cutello andA.Policriti Set\u2010theoretic reduction of Hilbert's tenth problem. In: Proceedings of Logic in Computer Science \u203289 Springer Lecture Notes in Computer Science (to appear)."},{"key":"e_1_2_1_6_2","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.3160410107"},{"key":"e_1_2_1_7_2","volume-title":"Computable Set Theory","author":"Cantone D.","year":"1990"},{"key":"e_1_2_1_8_2","volume-title":"Theorie axiomatique des ensembles","author":"Krivine J. L.","year":"1969"},{"key":"e_1_2_1_9_2","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.3160410206"},{"key":"e_1_2_1_10_2","doi-asserted-by":"publisher","DOI":"10.2307\/2047565"},{"key":"e_1_2_1_11_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF00243810"},{"key":"e_1_2_1_12_2","doi-asserted-by":"publisher","DOI":"10.2307\/2275470"},{"key":"e_1_2_1_13_2","doi-asserted-by":"publisher","DOI":"10.2307\/2047726"},{"key":"e_1_2_1_14_2","article-title":"Undecidability results for restricted universally quantified formulae of set theory","author":"Parlamento F.","journal-title":"Communications Pure Appl. Math."}],"container-title":["Mathematical Logic Quarterly"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fmalq.19920380110","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/malq.19920380110","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,9,27]],"date-time":"2023-09-27T20:35:37Z","timestamp":1695846937000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/malq.19920380110"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1992,1]]},"references-count":13,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1992,1]]}},"alternative-id":["10.1002\/malq.19920380110"],"URL":"https:\/\/doi.org\/10.1002\/malq.19920380110","archive":["Portico"],"relation":{},"ISSN":["0942-5616","1521-3870"],"issn-type":[{"value":"0942-5616","type":"print"},{"value":"1521-3870","type":"electronic"}],"subject":[],"published":{"date-parts":[[1992,1]]}}}