Beschreibung:

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). Kristiania [Oslo]: Jacob Dybwad, 1920. 4 o. 36 pages. Original gray printed wrappers; boxed. FIRST EDITION. In this 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" Another way of phrasing the theorem would be to state that if a theory has a model then it has a countable model. "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] ... Skolem's result can be regarded as establishing that the well-formed formulas in Skolem normal form constitute a reduction class for quantification theory. More generally, formulas in prenex form [a special form that can be transformed into normal form] came to play a major role in the study of the reduction problem and of special cases of the decision problem" (Heijenoort 1967, 252). Skolem's paper is reprinted in Heijenoort, From Frege to Gödel: A Sourcebook in Mathematical Logic, 1967, 254-63. OOC 365.