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A Duration Calculus with Infinite Intervals
 In Fundamentals of Computation Theory, Horst Reichel (Ed.), pages 1641. LNCS 965
, 1995
"... Abstract. This paper introduces infinite intervals into the Duration Calculus [32]. The extended calculus defines a state duration over an infinite interval by a property which specifies the limit of the state duration over finite intervals, and excludes the description operator. Thus the calculus c ..."
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Cited by 13 (2 self)
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Abstract. This paper introduces infinite intervals into the Duration Calculus [32]. The extended calculus defines a state duration over an infinite interval by a property which specifies the limit of the state duration over finite intervals, and excludes the description operator. Thus the calculus can be established without involvement of unpleasant calculation of infinity. With limits of state durations, one can treat conventional liveness and fairness, and can also measure liveness and fairness through properties of limits. Including both finite and infinite intervals, the calculus can, in a simple manner, distinguish between terminating behaviour and nonterminating behaviour, and therefore directly specify and reason about sequentiality. 1
A formalization of the Ramified Type Theory
, 1994
"... In "Principia Mathematica " [17], B. Russell and A.N. Whitehead propose a type system for higher order logic. This system has become known under the name "ramified type theory". It was invented to avoid the paradoxes, which could be conducted from Frege's "Begriffschrift&quo ..."
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Cited by 3 (1 self)
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In "Principia Mathematica " [17], B. Russell and A.N. Whitehead propose a type system for higher order logic. This system has become known under the name "ramified type theory". It was invented to avoid the paradoxes, which could be conducted from Frege's "Begriffschrift" [7]. We give a formalization of the ramified type theory as described in the Principia Mathematica, trying to keep it as close as possible to the ideas of the Principia. As an alternative, distancing ourselves from the Principia, we express notions from the ramified type theory in a lambda calculus style, thus clarifying the type system of Russell and Whitehead in a contemporary setting. Both formalizations are inspired by current developments in research on type theory and typed lambda calculus; see e.g. [3]. In these formalizations, and also when defining "truth", we will need the notion of substitution. As substitution is not formally defined in the Principia, we have to define it ourselves. Finally, the reaction by Hilbert and Ackermann in [10] on the
Hermann Weyl’s Intuitionistic Mathematics. Dirk
"... It is common knowledge that for a short while Hermann Weyl joined Brouwer in his pursuit of a revision of mathematics according to intuitionistic principles. There is, however, little in the literature that sheds light on Weyl’s role, and in particular on Brouwer’s reaction to Weyl’s allegiance to t ..."
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Cited by 1 (0 self)
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It is common knowledge that for a short while Hermann Weyl joined Brouwer in his pursuit of a revision of mathematics according to intuitionistic principles. There is, however, little in the literature that sheds light on Weyl’s role, and in particular on Brouwer’s reaction to Weyl’s allegiance to the cause of intuitionism. This short episode certainly raises a number of questions: what made Weyl give up his own program, spelled out in “Das Kontinuum”, how come Weyl was so wellinformed about Brouwer’s new intuitionism, in what respect did Weyl’s intuitionism differ from Brouwer’s intuitionism, what did Brouwer think of Weyl’s views,........? To some of these questions at least partial answers can be put forward on the basis of some of the available correspondence and notes. The present paper will concentrate mostly on the historical issues of the intuitionistic episode in Weyl’s career. Weyl entered the foundational controversy with a bang in 1920 with his sensational paper “On the new foundational crisis in mathematics ” 1. He had already made a name for himself in the foundations of mathematics in 1918 with his monograph “The Continuum” [Weyl 1918] ; this contained in addition to a technical logical – mathematical construction of the continuum, a fairly extensive discussion of the shortcomings of the traditional construction of the continuum on the basis of arbitrary — and hence also impredicative — Dedekind cuts. This book did not cause much of a stir in mathematics, that is to say, it was ritually quoted in the literature but, probably, little understood. It had to wait for a proper appreciation until the phenomenon of impredicativity was better understood 2. The paper “On the new foundational crisis in mathematics ” had a totally different effect, it was the proverbial stone thrown into the quiet pond of mathematics. Weyl characterised it in retrospect with the somewhat apologetic words: Only with some hesitation I acknowledge these lectures, which reflect in their style, which was here and there really bombastic, the mood of excited times — the times immediately following the First World War. 3 Indeed, Weyl’s “New crisis ” reads as a manifesto to the mathematical community, it uses an evocative language with a good many explicit references to the political
Difficulties of the set of natural numbers
"... In this article some difficulties are deduced from the set of natural numbers. The demonstrated difficulties suggest that if the set of natural numbers exists it would conflict with the axiom of regularity. As a result, we have the conclusion that the class of natural numbers is not a set but a prop ..."
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In this article some difficulties are deduced from the set of natural numbers. The demonstrated difficulties suggest that if the set of natural numbers exists it would conflict with the axiom of regularity. As a result, we have the conclusion that the class of natural numbers is not a set but a proper class.
An implication of Gödel’s incompleteness theorem
, 2009
"... A proof of Gödel’s incompleteness theorem is given. With this new proof a transfinite extension of Gödel’s theorem is considered. It is shown that if one assumes the set theory ZFC on the meta level as well as on the object level, a contradiction arises. The cause is shown to be the implicit identif ..."
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A proof of Gödel’s incompleteness theorem is given. With this new proof a transfinite extension of Gödel’s theorem is considered. It is shown that if one assumes the set theory ZFC on the meta level as well as on the object level, a contradiction arises. The cause is shown to be the implicit identification of the meta level and the object level hidden behind the Gödel numbering. An implication of these considerations is stated.