{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,7,15]],"date-time":"2024-07-15T13:11:00Z","timestamp":1721049060707},"reference-count":32,"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":4394,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2002,3]]},"abstract":"Abstract<\/jats:title>We confirm a conjecture, about neat embeddings of cylindric algebras, made in 1969 by J. D. Monk, and a later conjecture by Maddux about relation algebras obtained from cylindric algebras. These results in algebraic logic have the following consequence for predicate logic: for every finite cardinal \u03b1<\/jats:italic> \u2265 3 there is a logically valid sentence X<\/jats:italic>, in a first-order language \u2112<\/jats:italic> with equality and exactly one nonlogical binary relation symbol E<\/jats:italic><\/jats:bold>, such that X<\/jats:italic> contains only 3 variables (each of which may occur arbitrarily many times), X<\/jats:italic> has a proof containing exactly \u03b1<\/jats:italic> + 1 variables, but X<\/jats:italic> has no proof containing only \u03b1<\/jats:italic> variables. This solves a problem posed by Tarski and Givant in 1987.<\/jats:p>","DOI":"10.2178\/jsl\/1190150037","type":"journal-article","created":{"date-parts":[[2007,12,13]],"date-time":"2007-12-13T19:12:10Z","timestamp":1197573130000},"page":"197-213","source":"Crossref","is-referenced-by-count":19,"title":["Relation algebra reducts of cylindric algebras and an application to proof theory"],"prefix":"10.1017","volume":"67","author":[{"given":"Robin","family":"Hirsch","sequence":"first","affiliation":[]},{"given":"Ian","family":"Hodkinson","sequence":"additional","affiliation":[]},{"given":"Roger D.","family":"Maddux","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200009920_ref027","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1971-0276063-5"},{"key":"S0022481200009920_ref017","doi-asserted-by":"publisher","DOI":"10.2307\/1969375"},{"key":"S0022481200009920_ref014","volume-title":"Annals of Pure and Applied Logic","author":"Hirsch"},{"key":"S0022481200009920_ref007","doi-asserted-by":"publisher","DOI":"10.4153\/CJM-1955-001-4"},{"key":"S0022481200009920_ref006","unstructured":"Givant S. , Tarski's development of logic and mathematics based on the calculus of relations, In [3], pp. 189\u2013215."},{"key":"S0022481200009920_ref005","first-page":"341","article-title":"Distributive and modular laws in the arithmetic of relation algebras","volume":"1","author":"Chin","year":"1951","journal-title":"University of California Publications in Mathematics, New Series"},{"key":"S0022481200009920_ref031","volume-title":"A formalization of set theory without variables","volume":"41","author":"Tarski","year":"1987"},{"key":"S0022481200009920_ref022","unstructured":"Maddux R. D. , Introductory course on relation algebras, finite-dimensional cylindric algebras, and their interconnections, In [3], pp. 361\u2013392."},{"key":"S0022481200009920_ref015","first-page":"344","volume":"34","author":"Johnson","year":"1969","journal-title":"Nonfinitizability of classes of representable polyadic algebras"},{"key":"S0022481200009920_ref021","first-page":"951","volume":"54","author":"Maddux","year":"1989","journal-title":"Non-finite axiomatizability results for cylindric and relation algebras"},{"key":"S0022481200009920_ref025","unstructured":"Monk J. D. , Studies in cylindric algebra, Ph.D. thesis , University of California, Berkeley, 1961."},{"key":"S0022481200009920_ref003","volume-title":"Algebraic logic","author":"Andr\u00e9ka","year":"1991"},{"key":"S0022481200009920_ref016","first-page":"576","volume":"38","author":"Johnson","year":"1973","journal-title":"Axiom systems for logic with finitely many variables"},{"key":"S0022481200009920_ref012","volume-title":"Cylindric algebras, Part II","author":"Henkin","year":"1985"},{"key":"S0022481200009920_ref009","first-page":"86","article-title":"Relativization with respect to formulas and its use in proofs of independence","volume":"20","author":"Henkin","year":"1968","journal-title":"Compositio Mathematica"},{"key":"S0022481200009920_ref013","first-page":"83","volume-title":"Lattice theory, proceedings of symposia in pure mathematics","volume":"2","author":"Henkin","year":"1961"},{"key":"S0022481200009920_ref032","first-page":"327","volume-title":"Algebraic methods in logic and in computer science","volume":"28","author":"Thompson","year":"1993"},{"key":"S0022481200009920_ref018","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1307\/mmj\/1028998510","article-title":"Relation algebras and projective geometries","volume":"8","author":"Lyndon","year":"1961","journal-title":"Michigan Mathematics Journal"},{"key":"S0022481200009920_ref023","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1991-1033959-7"},{"key":"S0022481200009920_ref011","volume-title":"Cylindric algebras, Part I","author":"Henkin","year":"1971"},{"key":"S0022481200009920_ref004","volume-title":"Handbook of philosophical logic","author":"Andr\u00e9ka"},{"key":"S0022481200009920_ref002","doi-asserted-by":"publisher","DOI":"10.1016\/S0168-0072(97)00027-4"},{"key":"S0022481200009920_ref024","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1307\/mmj\/1029000477","article-title":"The representation of integral relation algebras","volume":"17","author":"McKenzie","year":"1970","journal-title":"Michigan Mathematical Journal"},{"key":"S0022481200009920_ref026","first-page":"331","volume":"34","author":"Monk","year":"1969","journal-title":"Nonfinitizability of classes of representable cylindric algebras"},{"key":"S0022481200009920_ref020","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0098465"},{"key":"S0022481200009920_ref028","doi-asserted-by":"publisher","DOI":"10.1093\/jigpal\/5.4.575"},{"key":"S0022481200009920_ref030","doi-asserted-by":"publisher","DOI":"10.1007\/BF01972461"},{"key":"S0022481200009920_ref001","volume-title":"Finite axiomatizability of SNrnCAn+1 and non-finite axiomatizability of SNrnCAn+2","author":"Andr\u00e9ka","year":"1990"},{"key":"S0022481200009920_ref010","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(08)71537-4"},{"key":"S0022481200009920_ref029","unstructured":"Simon A. , Nonrepresentable algebras of relations, Ph.D. thesis , Math. 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D. , Topics in relation algebra, Ph.D. thesis , University of California, Berkeley, 1978."},{"key":"S0022481200009920_ref008","volume-title":"Logic systems containing only a finite number of symbols","volume":"21","author":"Henkin","year":"1967"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200009920","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,7]],"date-time":"2019-05-07T01:34:15Z","timestamp":1557192855000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200009920\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2002,3]]},"references-count":32,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2002,3]]}},"alternative-id":["S0022481200009920"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1190150037","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2002,3]]}}}