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Perfect Skolem sets
"... A Skolem sequence is a sequence s1, s2,..., s2n (where si ∈ A = {1... n}), each si occurs exactly twice in the sequence and the two occurrences are exactly si positions apart. A set A that can be used to construct Skolem sequences is called a Skolem set. The problem of deciding which sets of the for ..."
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of the form A = {1... n} are Skolem sets was solved by Thoralf Skolem in the late 1950’s. We study the natural generalization where A is allowed to be any set of n positive integers. We give necessary conditions for the existence of Skolem sets of this generalized form. We conjecture these necessary
N Pcompleteness of generalized multi Skolem sequences
"... A Skolem sequence is a sequence a1,a2,...,a2n (where ai ∈ A = {1,...,n}), each ai occurs exactly twice in the sequence and the two occurrences are exactly ai positions apart. A set A that can be used to construct Skolem sequences is called a Skolem set. The existence question of deciding which sets ..."
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of the form A = {1,...,n} are Skolem sets was solved by Thoralf Skolem [6] in 1957. Many generalizations of Skolem sequences have been studied. In this paper we prove that the existence question for generalized multi Skolem sequences is N Pcomplete. This can be seen as an upper bound on how far
The emergence of firstorder logic
 University of Minnesota Press, Minneapolis
, 1988
"... To most mathematical logicians working in the 1980s, firstorder logic is the proper and natural framework for mathematics. Yet it was not always so. In 1923, when a young Norwegian mathematician named Thoralf Skolem argued that set theory should be based on firstorder logic, it was ..."
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To most mathematical logicians working in the 1980s, firstorder logic is the proper and natural framework for mathematics. Yet it was not always so. In 1923, when a young Norwegian mathematician named Thoralf Skolem argued that set theory should be based on firstorder logic, it was
PIONEER OF COMPUTATIONAL LOGIC
"... In a way, the title is misleading. Thoralf Skolem did not—as far as I know—use a computer or do any serious computations in logic. He was one of a handful of men who founded modern logic. But in his work he always stressed the computational aspects, perhaps more so ..."
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In a way, the title is misleading. Thoralf Skolem did not—as far as I know—use a computer or do any serious computations in logic. He was one of a handful of men who founded modern logic. But in his work he always stressed the computational aspects, perhaps more so
Philma: “bookreview ” — 2005/1/21 — 10:30 — page 91 — #14 CRITICAL STUDIES/BOOK REVIEWS 91 What Did Löwenheim Prove? Calixto Badesa. The Birth of Model Theory: Löwenheim’s Theorem
"... When we encounter a theorem with a composite name, like HeineBorel, CantorBendixson, or LöwenheimSkolem, we are curious to know what the particular contribution to it of each author actually was. The obvious guess is an alternative: either the first author provided a deficient or incomplete proo ..."
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plete proof, or else the second author generalized the original theorem. As regards the LöwenheimSkolem theorem, both things are the case. The theorem was first proved in 1915 by Leopold Löwenheim (1878–1957), and then reproved and generalized by Thoralf Skolem (1887–1963) in 1920, in 1922, and again in 1929
History of Constructivism in the 20th Century
"... notions, such as `constructive proof', `arbitrary numbertheoretic function ' are rejected. Statements involving quantifiers are finitistically interpreted in terms of quantifierfree statements. Thus an existential statement 9xAx is regarded as a partial communication, to be supplemented ..."
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by providing an x which satisfies A. Establishing :8xAx finitistically means: providing a particular x such that Ax is false. In this century, T. Skolem 4 was the first to contribute substantially to finitist 4 Thoralf Skolem 18871963 History of constructivism in the 20th century 3 mathematics; he