Thoralf Skolem (21 results)

Condition: New.

Hardback or Cased Book. Condition: New. Abstract Set Theory; 8 0.67. Book.

Soft cover. Condition: Very Good. ] Trade pb, 6x9 inches, white titling on red-brown front cover, vi + 70 pages, VG+. Very nice overall condition with only very light signs of any wear. Unprinted red-brown spine color shifted to gray. Professor's last name top edge first interior page (title page) o/w crisp clean throughout interior. RWR5 Math Mathematics.

Leatherbound. Condition: NEW. Leatherbound edition. Condition: New. Leather Binding on Spine and Corners with Golden leaf printing on spine. Bound in genuine leather with Satin ribbon page markers and Spine with raised gilt bands. A perfect gift for your loved ones. Reprinted from 1917 edition. NO changes have been made to the original text. This is NOT a retyped or an ocr'd reprint. Illustrations, Index, if any, are included in black and white. Each page is checked manually before printing. As this print on demand book is reprinted from a very old book, there could be some missing or flawed pages, but we always try to make the book as complete as possible. Fold-outs, if any, are not part of the book. If the original book was published in multiple volumes then this reprint is of only one volume, not the whole set. IF YOU WISH TO ORDER PARTICULAR VOLUME OR ALL THE VOLUMES YOU CAN CONTACT US. Resized as per current standards. Sewing binding for longer life, where the book block is actually sewn (smythe sewn/section sewn) with thread before binding which results in a more durable type of binding. Pages: 424 Language: German Volume 1917 Pages: 424 Volume 1917.

Leatherbound. Condition: NEW. Leatherbound edition. Condition: New. Leather Binding on Spine and Corners with Golden leaf printing on spine. Bound in genuine leather with Satin ribbon page markers and Spine with raised gilt bands. A perfect gift for your loved ones. Reprinted from 1922 edition. NO changes have been made to the original text. This is NOT a retyped or an ocr'd reprint. Illustrations, Index, if any, are included in black and white. Each page is checked manually before printing. As this print on demand book is reprinted from a very old book, there could be some missing or flawed pages, but we always try to make the book as complete as possible. Fold-outs, if any, are not part of the book. If the original book was published in multiple volumes then this reprint is of only one volume, not the whole set. IF YOU WISH TO ORDER PARTICULAR VOLUME OR ALL THE VOLUMES YOU CAN CONTACT US. Resized as per current standards. Sewing binding for longer life, where the book block is actually sewn (smythe sewn/section sewn) with thread before binding which results in a more durable type of binding. Pages: 868 Language: German Volume 1921 no.9-20 Pages: 868 Volume 1921 no.9-20.

Nach- oder Neudruck der Ausgabe J. Springer, Berlin, 1938, 8°, 130 S., dunkelroter original Leineneinband mit goldener Rückenbeschriftung. sehr schönes, sauberes Exemplar ohne nennenswerte Mängel Aufgrund des Zustandes vermuten wir, das es sich hier um einen Neudruck handeln muss, da dieser Band ansonsten unter Licht- und Luftabschluß im Hochvakuum aufbewahrt worden sein muss. Einen Hinweis dazu enthält das Buch aber nicht.

Condition: good. A good reading copy. May contain markings or be a withdrawn library copy.

First edition, first printing, collecting the lectures delivered at the symposium on "mathematical interpretation of formal systems", at Amsterdam in September 1954. The contributors are Th. Skolem, G. Hasenjaeger, G. Kreisel, A. Robinson, Hao Wang, L. Henkin, and J. Los. Octavo. Original yellow wrappers, lettered in black and blue. With dust jacket. Tiny nick at head of front joint, minor peripheral bumping, a near-fine copy in very good jacket, spine panel darkened with chip at head, light soiling.

Soft cover. Condition: Very Good. 1st Edition. 37 pp. Original wrappers. Spine repaired (see photos). Signature of former owner Samuel Skulsky on title page. Very Good. First Edition. "Skolem's second paper in logic, dated 26 May 1917 and published in Norway in 1919. The work contains important early results on lattice theory, including the following: 1. A proof of the mutual independence of all the lattice axioms through graphical models, some of them infinite. 2. The same for distributive lattices. In particular, the five-element lattice that is a counterexample to distributivity is given. 3. A method for deciding if an atomic formula is derivable from at most two atomic formulas by the axioms for a distributive lattice. Skolem's paper also introduces the notion of a Heyting algebra. . . . It is quite astonishing that Skolem had found an algebraic axiomatization of intuitionistic propositional logic well before its basic principles were definitively clarified by Heyting's axiomatization (1930). Of his motivations for introducing the algebraic axiomatization, what he called 'class rings' (Klassenringe), Skolem writes that they are 'a natural continuation and generalization of the groups of the identity calculus' (1919, p. 78). Whatever the motivations, Skolem had invented the notion of a Heyting algebra directly from lattice theory, through a symbolic maximal solution of the defining inequality of the arrow. . . . Skolem's work was appreciated by Dana Scott who wrote, in a review of a late lattice-theoretic paper Skolem (1958): 'In a remarkable paper published in 1919 . . . the author studied several questions about distributive lattices in a completely abstract and general form. At that time, however, he did not use the terminology we know today. . . . Hence, his work escaped the notice that it actually deserves' " (Jan van Plato, "In the Shadows of the Löwenheim-Skolem Theorem: Early Combinatorial Analyses of Mathematical Proofs," Bulletin of Symbolic Logic, Vol. 13, no. 2, June 2007).

1. Einige Bemerkungen zu der Abhandlung von E. Zermelo: "Über die Definitheit in der Axiomatik." Offprint from Fundamenta mathematicae 15 (1930). 337-341pp. Original printed wrappers. 2. Über die symmetrisch allgemein Lösungen im Klassenkalkul. Offprint from Fundamenta mathematicae 18 (1931). 61-76pp. Original printed wrappers. 3. Ein kombinatorischer Satz mit Anwendung auf ein logisches Entscheidungsproblem. Offprint from Fundamenta mathematicae 20 (1933). 254-261pp. Original printed wrappers. 4. Über die Nicht-charakterisierbarkeit der Zahlenreihe mittels endlich oder abzählbar unendlich vieler Aussagen mit ausschliesslich Zahlenvariablen. Offprint from Fundamenta mathematicae 23 (1934). 150-161pp. Original printed wrappers. 5. Einige Sätze über p-adisch Potenzreihen mit Anwendung auf gewisse exponentielle Gleichungen. Offprint from Mathematische Annalen 111 (1935). 399-424pp. Original printed wrappers. 6. Über gewisse "Verbände" oder "Lattices." Offprint from Det norske Videnskaps-Akademie i Oslo, Matematisk-naturvidenskapelig klasse (1936). 16pp. Original printed wrappers. 7. Einige Reduktionen des Entscheidungsproblems. Offprint from Det norske videnskaps-akademie i Oslo, matematisk-naturvidenskapelig klasse (1936). 17pp. Original printed wrappers. 8. Ein Satz über ganzwertige Polynome. Offprint from Det kongelige norske videnskabers selskab 9 (1936). 111-113pp. Original printed wrappers. 9. Über die Lösbarkeit gewisser linearer Gleichungen im Bereiche der ganzwertigen Polynome. Offprint from Det kongelige norske videnskabers selskab 9 (1936). 134-137pp. Original printed wrappers. 10. Sätze über ganzwertige Polynome. Offprint from Det kongelige norske videnskabers selskab 10 (1937). 12-15pp. Original printed wrappers. 11. Zwei Sätze über kubische Kongruenzen. Offprint from Det kongelige norske videnskabers selskab 10 (1937). 89-92pp. Original printed wrappers. 12. Über eine Eigenschaft der Menge aller grössten gemeinsamen Teiler von Polynomen für ganze Werte der Variablen. Offprint from Det kongelige norske videnskabers selskab 11 (1938). 64-67pp. Original printed wrappers. 13. Über die Lösbarkeit der Gleichung f1 (x) F1 (x) + . . . . + fn (x) Fn (x) = 1, wo f1, . . . , fn gegebene ganzzahlige Polynome sind, in ganzzahligen Polynomen F1, . . . , Fn. Offprint from Det kongelige norske videnskabers selskab 12 (1939). 4pp. Original printed wrappers. 14. Eine Bemerkung über die Induktionsschemata in der rekursiven Zahlentheorie. Offprint from Monatsheften für Mathematik und Physik 48 (1939). 268-276pp. Original printed wrappers. 15. Einfacher Beweis der Unmöglichkeit eines allgemeinen Lösungsverfahrens für arithmetische Probleme. Offprint from Det kongelige norske videnskabers selskab 13 (1940).4pp. Original printed wrappers. 16. Die Anzahl der Wurzeln der Kongruenz x3 + ax + b 0 (mod p) für die verschiedenen Paare a, b. Offprint from Det kongelige norske videnskabers selskab 14 (1941). 161-164pp. Original printed wrappers. 17. A note on recursive arithmetic. Offprint from Det kongelige norske videnskabers selskab 17 (1943). 107-109pp. Original printed wrappers. 18. Remarks on recursive functions and relations. Offprint from Det kongelige norske videnskabers selskab 17 (1944). 89-92pp. Original printed wrappers. 19. Some remarks on the comparison between recursive functions. Offprint from Det kongelige norske videnskabers selskab 17 (1944). 126-129pp. Original printed wrappers. 20. Some remarks on recursive arithmetic. Offprint from Det kongelige norske videnskabers selskab 17 (1944). 103-106pp. Original printed wrappers. 21. Über Nebenkörper und Nebenringe. Offprint from Det norske Videnskaps-Akademie i Oslo, Matematisk-naturvidenskapelig klasse (1944). 70pp. Original printed wrappers. 22. A theorem on the equation 2 - 2 = 1 where , , are integers in an imaginary quadratic field. Offprint from Det norske Videnskaps-Akademie i Oslo, Matematisk-naturvidenskapelig klasse (1945). 13pp. Original printed wrappers. 23. A remark on the equation 2 - 2 = 1, > 0, , , . . < 0, where , , belong to a total real number field. Offprint from Det norske Videnskaps-Akademie i Oslo, Matematisk-naturvidenskapelig klasse (1945). 15pp. Original printed wrappers. 24. On the prime divisors of the values of certain functions. Offprint from Det kongelige norske videnskabers selskab 18 (1945). 75-78pp. Original printed wrappers. 25. On certain exponential equations. Offprint from Det kongelige norske videnskabers selskab 18 (1945). 71-74pp. Original printed wrappers. 26. Some considerations concerning recursive functions. Offprint from Mathematica scandinavica 1 (1953). 213-221pp. Without wrappers as issued. Creased vertically. .

Skolem, Thoralf (1887-1963). Logisch-kombinatorische Untersuchungen über der Erfüllbarkeit und Beweisbarkeit mathematischen Sätze nebst einem Theoreme über dichte Mengen. Offprint from Skrifter utgit av Videnskabsselskapet i Kristiania, I. Matematisk-naturvidenskabelig klasse, 4 (1920). 36pp. 274 x 182 mm. Later wrappers, but very good. First Edition, Offprint Issue. "Many logicians would agree that Skolem and Gödel are the two greatest logicians of the [20th] century" (Wang, p. 119). In the present paper, Skolem "proved the theorem which is now known as the Löwenheim-Skolem theorem. Briefly, this states that if a finite or countable infinite set of sentences formalized within a first order predicate calculus is satisfiable (or, in other terminology, has a model), then the sentences are satisfiable within a countable domain" (Fenstad, p. 102). Another way of phrasing the theorem would be to state that if a theory has a model, then it has a countable model. The Löwenheim-Skolem theorem is one of the foundations of automated theorem proving. "The key work for automated deduction was that of Skolem. He carried out a systematic study of the problem of the existence of an interpretation which will satisfy a given formula of the predicate calculus, or, as one says, whether the given formula is satisfiable" (Davis, p. 9). Skolem was an exception to the rule that mathematicians do their best work when youngâ "his first important paper dates from 1919, when he was thirty-two years old, and he continued to publish significant work until the end of his life. The papers that he published from 1920 on "gave him instant recognition as one of the foremost among logicians" (Fenstad, p. 101). M. Davis, "The prehistory and early history of automated deduction," Automation of Reasoning: Classical Papers on Computational Logic (1983): 1-28. J. Fenstad, "Thoralf Albert Skolem 1887-1963: A brief biographical sketch," Nordic Journal of Philosophical Logic 1 (1996): 99-106. Heijenoort, From Frege to Gödel, pp. 252-263 (reprinting Skolem's paper). Origins of Cyberspace 365. Wang, "Skolem and Gödel," Nordic Journal of Philosophical Logic 1 (1996): 119-132. .

Soft cover. Condition: Near Fine. 1st Edition. 36 pp. Original printed wrappers. UNOPENED COPY. Near Fine. First Edition. Löwenheim-Skolem Theorem. "Many logicians would agree that Skolem and Gödel are the two greatest logicians of the [20th] century" (Wang, "Skolem and Gödel", Nordic Journal of Philosophical Logic, Vol. 1, no. 2, pp. 119-132). "The first result presented in the paper is that every well-formed formula of the first-order predicate calculus has what is now known as a Skolem normal form for satisfiability. . . . Skolem normal forms have since become one of the logician's standard tools. These forms were used, in particular, by Gödel in his proof of the completeness of quantification theory (1930). . . . After having introduced the normal form of a formula, Skolem offers (Theorem 2) a new proof of Löwenheim's theorem. . . . He is able to fill a serious gap left open by Löwenheim in his proof. Skolem notes the use of the axiom of choice in his proof. . .(Van Heijenoort, From Frege to Gödel, p. 252; English translation (of §1) Logico-Combinatorial Investigations in the Satisfiability or Provability of Mathematical Propositions: A Simplified Proof of a Theorem by L. Löwenheim and Generalizations of the Theorem in Van Heijenoort, pp. 254-63). "It seems clear that Godel read at least the statements of theorems and definitions in Löwenheim's paper, and in Skolem's 1920 paper as well. In the published version of his thesis (1930), Gödel cites Skolem (1920) explicitly: 'An analogous procedure was used by Skolem (1920) in proving Löwenheim's theorem' (Gödel 1930, pp. 108-109). . . . Skolem studied at Göttingen in the winter of 1915-1916. We do not know whether he first learned of Löwenheim's paper at Göttingen, or whether he simply read it in the Mathematische Annalen. Skolem's first paper re-proving Löwenheim's theorem (1920) introduced the notion of first-order equations and dropped the relative sum and product notation that Löwenheim had adopted from Schröder, and which originated with Peirce. Skolem's 1920 proof was thus a simplified version of Löwenheim's original proof using algebraic notations and the axiom of choice. Skolem did not claim that Löwenheim's original proof was wrong or incomplete; he only said he was giving a simpler and clearer proof. . . . Stated in the language of the calculus of relatives, Löwenheim's celebrated theorem about first-order logic simply fell out of what was for him an obvious technical generalization of something that was more interesting in the calculus of relatives, that is, a technique that enables him to say what the cardinalities of universes can be from the analysis of the form of an equation. It was left to Skolem to extract the essence of Löwenheim's theorem and state it in the language of first-order logic. . . . [A]lthough Löwenheim's paper was published in 1915 in the premier mathematical journal in the world, it was not until Skolem's paper of 1920 that Löwenheim's theorem received any attention. . . . Skolem's 1920 paper . . . begins with a theorem known today as the Skolem normal form. . . . Skolem's 1920 proof of Löwenheim's theorem uses the axiom of choice. It is quite simple. It is also not Löwenheim's proof, although the ideas are present in Löwenheim in a less clear form and in a longer proof. . ." (Brady, From Peirce to Skolem: A Neglected Chapter in the History of Logic pp. 3-4, 170-71, 197-98). OCLC locates copies in these 5 US libraries: Chicago, Iowa, Johns Hopkins, Stanford, Washington.