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A New Approach of Innovative Design: an Introduction to
 CK Theory,” Proceedings, International Conference on Engineering Design
, 2003
"... In this paper we introduce the main notions and first applications of a unified design theory. We call it “CK theory ” because it stands that a formal distinction between spaces of “Concepts ” (C) and space of “Knowledge ” (K) is a condition for design. This distinction has key properties: i) it ..."
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In this paper we introduce the main notions and first applications of a unified design theory. We call it “CK theory ” because it stands that a formal distinction between spaces of “Concepts ” (C) and space of “Knowledge ” (K) is a condition for design. This distinction has key properties: i) it
The Mathematical Import Of Zermelo's WellOrdering Theorem
 Bull. Symbolic Logic
, 1997
"... this paper, the seminal results of set theory are woven together in terms of a unifying mathematical motif, one whose transmutations serve to illuminate the historical development of the subject. The motif is foreshadowed in Cantor's diagonal proof, and emerges in the interstices of the inclusi ..."
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this paper, the seminal results of set theory are woven together in terms of a unifying mathematical motif, one whose transmutations serve to illuminate the historical development of the subject. The motif is foreshadowed in Cantor's diagonal proof, and emerges in the interstices of the inclusion vs. membership distinction, a distinction only clarified at the turn of this century, remarkable though this may seem. Russell runs with this distinction, but is quickly caught on the horns of his wellknown paradox, an early expression of our motif. The motif becomes fully manifest through the study of functions f :
Ordinals and Interactive Programs
, 2000
"... The work reported in this thesis arises from the old idea, going back to the origins of constructive logic, that a proof is fundamentally a kind of program. If proofs can be ..."
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Cited by 5 (2 self)
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The work reported in this thesis arises from the old idea, going back to the origins of constructive logic, that a proof is fundamentally a kind of program. If proofs can be
Tarski’s Intuitive Notion of Set
"... Abstract. Tarski did research on set theory and also used set theory in many of his emblematic writings. Yet his notion of set from the philosophical viewpoint was almost unknown. By studying mostly the posthumously published evidence, his still unpublished materials, and the testimonies of some of ..."
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Abstract. Tarski did research on set theory and also used set theory in many of his emblematic writings. Yet his notion of set from the philosophical viewpoint was almost unknown. By studying mostly the posthumously published evidence, his still unpublished materials, and the testimonies of some of his collaborators, I try to offer here a first, global picture of that intuitive notion, together with a philosophical interpretation of it. This is made by using several notions of universal languages as framework, and by taking into consideration the evolution of Tarski’s thoughts about set theory and its relationship with logic and mathematics. As a result, his difficulties to reconcile nominalism and methodological Platonism are precisely located, described and much better understood. “I represent this very rude kind of antiPlatonism, one thing which I could describe as materialism, or nominalism with some materialistic taint, and it is very difficult for a man to live his whole life with this philosophical attitude, especially if he is a mathematician, especially if for some reasons he has a hobby which is called set theory, and worse –very difficult” (Tarski, Chicago, 1965) Tarski made important contributions to set theory, especially in the first years of his long and highly productive career. Also, it is usually accepted that set theory was the main instrument used by Tarski in his most significant contributions which had philosophical implications and presuppositions. In this connection we may mention the four definitions which are usually cited as supplying some sort of “conceptual analysis”, both methodologically (the first one) and from the point of view of the results obtained (the rest): (i) definable sets of real numbers; (ii) truth; (iii) logical consequence and (iv) logical notions. So we could reasonably conclude that for Tarski set theory was reliable as a working instrument, then presumably as a conceptual ground. As we shall see, there are some signs that the reason for this preference might have been its simple ontological structure as a theory, at least excluding the upper levels, the highest infinite. Nevertheless, very little was known about Tarski’s conception of set theory from the philosophical viewpoint, apart from some comments he made to his closest friends and collaborators, and the conjectures which could perhaps be
CANTOR ET LES INFINIS
, 2009
"... En 1874 paraı̂t au Journal de Crelle une note de quatre pages ou ̀ Georg Cantor, alors age ́ de vingtneuf ans et jeune professeur a ̀ l’universite ́ de Halle, établit la dénombrabilite ́ de l’ensemble des nombres algébriques et la nondénombrablite ́ de l’ensemble des nombres réels. Cet articl ..."
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En 1874 paraı̂t au Journal de Crelle une note de quatre pages ou ̀ Georg Cantor, alors age ́ de vingtneuf ans et jeune professeur a ̀ l’universite ́ de Halle, établit la dénombrabilite ́ de l’ensemble des nombres algébriques et la nondénombrablite ́ de l’ensemble des nombres réels. Cet article est révolutionnaire car, pour la première fois, l’infini est considére ́ non plus comme une limite inatteignable mais comme un possible objet d’investigation. L’héritage de ce travail est extraordinaire: non seulement il marque la naissance de la théorie des ensembles — en fait une théorie de l’infini — mais il contient déja ̀ en germe le problème du continu qui a occupe ́ toute la fin de la vie de Cantor et a éte ́ et continue d’être le moteur du développement de cette théorie. Un temps objet d’une fascination déraisonnable reposant sur un malentendu, celleci est aujourd’hui largement méconnue, alors même qu’apparaissent les premiers signes d’une possible résolution du problème du continu pose ́ par Cantor. Ce texte présente le contexte et le contenu de l’article de Cantor, puis évoque deux des principaux développements qui en sont issus, a ̀ savoir la construction des ordinaux transfinis, avec l’amusante application aux suites de Goodstein, et le problème du continu, y compris les contresens souvent rencontrés sur la signification des résultats de Gödel et Cohen, ainsi que les résultats récents de Woodin qui laissent entrevoir ce que pourrait être une solution future. 1. Une petite note et deux résultats simples 1.1. L’auteur. Georg Cantor naı̂t en 1845 a ̀ SaintPetersbourg, d’une mère russe et d’un père homme d’affaires allemand, d’origine juive mais converti au protestantisme. Il passe ses premières années en Russie. La famille revient en Allemagne quand Georg a onze ans, d’abord a ̀ Wiesbaden puis à Francfort. Cantor fréquente le lycée de Darmstadt, ou ̀ ses dons en mathématiques sont remarqués, puis
Foundations of Regular Variation
"... The theory of regular variation is largely complete in one dimension, but is developed under regularity or smoothness assumptions. For functions of a real variable, Lebesgue measurability suffices, and so does having the property of Baire. We find here that the preceding two properties have two kin ..."
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The theory of regular variation is largely complete in one dimension, but is developed under regularity or smoothness assumptions. For functions of a real variable, Lebesgue measurability suffices, and so does having the property of Baire. We find here that the preceding two properties have two kinds of common generalization, both of a combinatorial nature; one is exemplified by ‘containment up to translation of subsequences’, the other, drawn from descriptive set theory, requires nonemptiness of a Souslin ∆ 1 2set. All of our generalizations are equivalent to the uniform convergence property.
Abstract—This is a brief overview of the basic techniques of Boolean valued analysis.
, 2006
"... The term “Boolean valued analysis ” appeared within the realm of mathematical logic. It was Takeuti, a renowned expert in proof theory, who introduced the term. Takeuti defined Boolean valued analysis in [18, p. 1] as “an application of Scott–Solovay’s Boolean valued models of set theory to analysis ..."
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The term “Boolean valued analysis ” appeared within the realm of mathematical logic. It was Takeuti, a renowned expert in proof theory, who introduced the term. Takeuti defined Boolean valued analysis in [18, p. 1] as “an application of Scott–Solovay’s Boolean valued models of set theory to analysis.” Vopěnka invented similar models at the same time. That is how the question of the title receives
WHAT IS BOOLEAN VALUED ANALYSIS?
, 2006
"... Abstract. This is a brief overview of the basic techniques of Boolean valued analysis. 1. ..."
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Abstract. This is a brief overview of the basic techniques of Boolean valued analysis. 1.