{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,19]],"date-time":"2026-06-19T09:00:19Z","timestamp":1781859619595,"version":"3.54.5"},"reference-count":54,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2014,1,15]],"date-time":"2014-01-15T00:00:00Z","timestamp":1389744000000},"content-version":"unspecified","delay-in-days":2420,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Bull. symb. log."],"published-print":{"date-parts":[[2007,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>The L\u00f6wenheim\u2013Skolem theorem was published in Skolem's long paper of 1920, with the first section dedicated to the theorem. The second section of the paper contains a proof-theoretical analysis of derivations in lattice theory. The main result, otherwise believed to have been established in the late 1980s, was a polynomial-time decision algorithm for these derivations. Skolem did not develop any notation for the representation of derivations, which makes the proofs of his results hard to follow. Such a formal notation is given here by which these proofs become transparent. A third section of Skolem's paper gives an analysis for derivations in plane projective geometry. To clear a gap in Skolem's result, a new conservativity property is shown for projective geometry, to the effect that a proper use of the axiom that gives the uniqueness of connecting lines and intersection points requires a conclusion with proper cases (logically, a disjunction in a positive part) to be proved.<\/jats:p><jats:p>The forgotten parts of Skolem's first paper on the L\u00f6wenheim\u2013Skolem theorem are the perhaps earliest combinatorial analyses of formal mathematical proofs, and at least the earliest analyses with profound results.<\/jats:p>","DOI":"10.2178\/bsl\/1185803805","type":"journal-article","created":{"date-parts":[[2007,12,13]],"date-time":"2007-12-13T14:55:58Z","timestamp":1197557758000},"page":"189-225","source":"Crossref","is-referenced-by-count":11,"title":["In the Shadows of the L\u00f6wenheim-Skolem Theorem: Early Combinatorial Analyses of Mathematical Proofs"],"prefix":"10.1017","volume":"13","author":[{"given":"Jan","family":"von Plato","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2014,1,15]]},"reference":[{"key":"S1079898600002213_ref054","unstructured":"Wang H. [1970], A survey of Skolem's work in logic, pp. 17\u201352 in Skolem (1970)."},{"key":"S1079898600002213_ref050","doi-asserted-by":"publisher","DOI":"10.1023\/A:1022997524909"},{"key":"S1079898600002213_ref013","first-page":"42","volume-title":"Sitzungsberichte der Preussischen Akademie von Wissenschaften, Physikalisch-mathematische Klasse","author":"Heyting","year":"1930"},{"key":"S1079898600002213_ref011","doi-asserted-by":"publisher","DOI":"10.1007\/BF01448897"},{"key":"S1079898600002213_ref008","doi-asserted-by":"publisher","DOI":"10.2307\/2964753"},{"key":"S1079898600002213_ref003","doi-asserted-by":"publisher","DOI":"10.2307\/2269018"},{"key":"S1079898600002213_ref043","first-page":"1","article-title":"Topological representations of distributive lattices and Brouwerian logics","volume":"67","author":"Stone","year":"1937","journal-title":"Casopis Pro Pestovani Matematiky a Fysiki Cast Matematicka"},{"key":"S1079898600002213_ref023","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511527340"},{"key":"S1079898600002213_ref009","doi-asserted-by":"publisher","DOI":"10.1007\/s00153-005-0321-z"},{"key":"S1079898600002213_ref012","volume-title":"Collected works V","author":"G\u00f6del","year":"2003"},{"key":"S1079898600002213_ref035","unstructured":"Skolem T. [1931b], \u00dcber die symmetrisch allgemeinen L\u00f6sungen im identischen Kalkul, as reprinted in Skolem 1970, pp. 307\u2013336."},{"key":"S1079898600002213_ref027","first-page":"000","article-title":"Review of Skolem","volume":"18","author":"Scott","year":"1958","journal-title":"Mathematical Reviews"},{"key":"S1079898600002213_ref024","doi-asserted-by":"publisher","DOI":"10.1007\/s000120200012"},{"key":"S1079898600002213_ref040","first-page":"169","article-title":"Consideraciones sobre los fundamentos de la matematica","volume":"12","author":"Skolem","year":"1952","journal-title":"Revista Matematica Hispano\u2013Americana"},{"key":"S1079898600002213_ref031","unstructured":"Skolem T. [1923], Einige Bemerkungen zur axiomatischen Begr\u00fcndung der Mengenlehre, as reprinted in Skolem 1970, pp. 137\u2013152."},{"key":"S1079898600002213_ref002","volume-title":"Solvable cases of the decision problem","author":"Ackermann","year":"1954"},{"key":"S1079898600002213_ref015","first-page":"1121","article-title":"Review of Skolem","volume":"48","author":"Lange-Nielsen","year":"1922","journal-title":"Jahresbericht \u00fcber die Fortschritte der Mathematik"},{"key":"S1079898600002213_ref007","doi-asserted-by":"publisher","DOI":"10.1016\/0890-5401(88)90048-X"},{"key":"S1079898600002213_ref026","first-page":"246","volume":"8","author":"Schroeder-Heister","year":"2002","journal-title":"Resolution and the origins of structural reasoning: early proof-theoretic ideas of Hertz and Gentzen"},{"key":"S1079898600002213_ref052","unstructured":"Plato J. von [2005], Combinatorial analysis of proofs in elementary geometry, ms."},{"key":"S1079898600002213_ref016","doi-asserted-by":"publisher","DOI":"10.1007\/BF01458217"},{"key":"S1079898600002213_ref028","first-page":"149","volume-title":"Proceedings of the 3rd Scandinavian mathematical congress","author":"Skolem","year":"1913"},{"key":"S1079898600002213_ref047","volume-title":"From Frege to G\u00f6del, A source book in mathematical logic","author":"Heijenoort","year":"1967"},{"key":"S1079898600002213_ref025","doi-asserted-by":"publisher","DOI":"10.1017\/S0960129504004244"},{"key":"S1079898600002213_ref038","volume-title":"Utvalgte kapitler av den matematiske logikk","author":"Skolem","year":"1936"},{"key":"S1079898600002213_ref036","first-page":"63","article-title":"Review of Gentzen","volume":"58","author":"Skolem","year":"1932","journal-title":"Jahresbericht \u00fcber die Fortschritte der Mathematik"},{"key":"S1079898600002213_ref032","unstructured":"Skolem T. [1928], \u00dcber die mathematische Logik, as reprinted in Skolem 1970, pp. 189\u2013206."},{"key":"S1079898600002213_ref045","first-page":"75","article-title":"On essential undecidability","volume":"14","author":"Tarski","year":"1949","journal-title":"The Journal of Symbolic Logic"},{"key":"S1079898600002213_ref019","volume-title":"Die Entstehung der Verbandstheorie","author":"Mehrtens","year":"1979"},{"key":"S1079898600002213_ref044","doi-asserted-by":"crossref","first-page":"103","DOI":"10.4064\/fm-31-1-103-134","article-title":"Der Aussagenkalk\u00fcl und die Topologie","volume":"31","author":"Tarski","year":"1938","journal-title":"Fundamenta Mathematicae"},{"key":"S1079898600002213_ref005","doi-asserted-by":"publisher","DOI":"10.1090\/coll\/025"},{"key":"S1079898600002213_ref033","unstructured":"Skolem T. [1929], \u00dcber einige Grundlagenfragen der Mathematik, as reprinted in Skolem 1970, pp. 227\u2013273."},{"key":"S1079898600002213_ref022","first-page":"418","volume":"4","author":"Negri","year":"1998","journal-title":"Cut elimination in the presence of axioms"},{"key":"S1079898600002213_ref017","doi-asserted-by":"publisher","DOI":"10.2307\/1969038"},{"key":"S1079898600002213_ref037","unstructured":"Skolem T. [1936], \u00dcber gewisse \u2018Verb\u00e4nde\u2019 oder \u2018Lattices\u2019, as reprinted in Skolem 1970, pp. 412\u2013424."},{"key":"S1079898600002213_ref042","volume-title":"Selected works in logic","author":"Skolem","year":"1970"},{"key":"S1079898600002213_ref030","unstructured":"Skolem T. [1920], Logisch-kombinatorische Untersuchungen \u00fcber die Erf\u00fcllbarkeit oder Beweisbarkeit mathematischer Satze, nebst einem Theoreme uber dichte Mengen, as reprinted in Skolem 1970, pp. 103\u2013136."},{"key":"S1079898600002213_ref034","doi-asserted-by":"publisher","DOI":"10.4064\/fm-18-1-61-76"},{"key":"S1079898600002213_ref039","unstructured":"Skolem T. [1941], Sur la port\u00e9e du th\u00e9or\u00e8me de L\u00f6wenheim\u2013Skolem, as reprinted in Skolem 1970, pp. 455\u2013482."},{"key":"S1079898600002213_ref020","first-page":"521","article-title":"Sequent calculus proof theory of intuitionistic apartness and order relations","volume":"38","author":"Negri","year":"1999","journal-title":"Archive for MathematicalLogic"},{"key":"S1079898600002213_ref004","first-page":"50","article-title":"Review of Skolem","volume":"2","author":"Birkhoff","year":"1937","journal-title":"The Journal of Symbolic Logic"},{"key":"S1079898600002213_ref010","doi-asserted-by":"publisher","DOI":"10.1090\/surv\/042"},{"key":"S1079898600002213_ref051","first-page":"427","volume-title":"Form, Zahl, Ordnung: Studien zur Wissenschafts- und Technikgeschichte","author":"Plato","year":"2004"},{"key":"S1079898600002213_ref053","article-title":"Gentzen s logic","volume":"5","author":"Plato","year":"2007","journal-title":"Handbook of the history and philosophy of logic"},{"key":"S1079898600002213_ref049","doi-asserted-by":"publisher","DOI":"10.1007\/s001530100091"},{"key":"S1079898600002213_ref048","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(95)00005-2"},{"key":"S1079898600002213_ref041","unstructured":"Skolem T. [1958], Remarks on the connection between intuitionistic logic and a certain class of lattices, as reprinted in Skolem 1970, pp. 639\u2013644."},{"key":"S1079898600002213_ref018","doi-asserted-by":"publisher","DOI":"10.2307\/2268135"},{"key":"S1079898600002213_ref021","doi-asserted-by":"publisher","DOI":"10.1007\/s001530100124"},{"key":"S1079898600002213_ref006","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19950410204"},{"key":"S1079898600002213_ref014","volume-title":"Untersuchungen zum Pr\u00e4dikatenkalk\u00fcl","author":"Ketonen","year":"1944"},{"key":"S1079898600002213_ref046","first-page":"382","article-title":"Schr\u00f6ders zweiter Beweis f\u00fcr die Unabh\u00e4ngigkeit der zweiten Subsumption des Distributivgesetzes im logischen Kalk\u00fcl","volume":"4","author":"Thiel","year":"1994","journal-title":"Modern Logic"},{"key":"S1079898600002213_ref001","first-page":"65","article-title":"Review of Skolem","volume":"58","author":"Ackermann","year":"1932","journal-title":"Jahresbericht \u00fcber die Fortschritte der Mathematik"},{"key":"S1079898600002213_ref029","unstructured":"Skolem T. [1919], Untersuchungen \u00fcber die Axiome des Klassenkalk\u00fcls und \u00fcber Produktations- und Summationsprobleme, welche gewisse Klassen von Aussagen betreffen, as reprinted in Skolem 1970, pp. 67\u2013101."}],"container-title":["Bulletin of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S1079898600002213","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,1]],"date-time":"2019-05-01T22:20:01Z","timestamp":1556749201000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S1079898600002213\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,6]]},"references-count":54,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2007,6]]}},"alternative-id":["S1079898600002213"],"URL":"https:\/\/doi.org\/10.2178\/bsl\/1185803805","relation":{},"ISSN":["1079-8986","1943-5894"],"issn-type":[{"value":"1079-8986","type":"print"},{"value":"1943-5894","type":"electronic"}],"subject":[],"published":{"date-parts":[[2007,6]]}}}