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Logisch-kombinatorische Untersuchungen über die Erfüllbarkeit oder Beweisbarkeit mathematischer Sätze nebst einem Theoreme über dichte Mengen. Videnskapsselskapets Skrifter. I. Mat.-naturv. klasse, 1920. no. 4. Utgit for Fridtjof Nansens Fond. Kristiania: in Kommission bei Jacob Dybwad, 1920 (Ink stamp: Printed in Norway). - 36 pp. Original wrappers. Wrappers remargined. Signature of former owner on front wrapper. 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 only 4 copies in US libraries: Chicago, Iowa, Stanford, Washington. [Attributes: First Edition; Soft Cover] [Bookseller: Scientia Books, ABAA]
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