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A Strongly Normalising CurryHoward Correspondence for IZF Set Theory
"... Abstract. We propose a method for realising the proofs of Intuitionistic ZermeloFraenkel set theory (IZF) by strongly normalising λterms. This method relies on the introduction of a Currystyle type theory extended with specific subtyping principles, which is then used as a lowlevel language to i ..."
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Abstract. We propose a method for realising the proofs of Intuitionistic ZermeloFraenkel set theory (IZF) by strongly normalising λterms. This method relies on the introduction of a Currystyle type theory extended with specific subtyping principles, which is then used as a lowlevel language to interpret IZF via a representation of sets as pointed graphs inspired by Aczel’s hyperset theory. As a consequence, we refine a classical result of Myhill and Friedman by showing how a strongly normalising λterm that computes a function of type N → N can be extracted from the proof of its existence in IZF. 1
2004a), Have Your Cake and Eat It Too: The Old Principal Principle Reconciled with the New
 Philosophy and Phenomenological Research
"... Abstract. David Lewis (1980) proposed the Principal Principle (PP) and a “reformulation ” which later on he called ‘OP ’ (Old Principle). Reacting to his belief that these principles run into trouble, Lewis (1994) concluded that they should be replaced with the New Principle (NP). This conclusion le ..."
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Abstract. David Lewis (1980) proposed the Principal Principle (PP) and a “reformulation ” which later on he called ‘OP ’ (Old Principle). Reacting to his belief that these principles run into trouble, Lewis (1994) concluded that they should be replaced with the New Principle (NP). This conclusion left Lewis uneasy, because he thought that an inverse form of NP is “quite messy”, whereas an inverse form of OP, namely the simple and intuitive PP, is “the key to our concept of chance”. I argue that, even if OP should be discarded, PP need not be. Moreover, far from being messy, an inverse form of NP is a simple and intuitive Conditional Principle (CP). Finally, both PP and CP are special cases of a General Principle (GP); it follows that so are PP and NP, which are thus compatible rather than competing.
Data models as constraint systems: A key to the semantic web. Research paper, submitted for publication
, 2007
"... This article illustrates how constraint logic programming can be used to express data models in rulebased languages, including those based on graph patternmatching or unification to drive rule application. This is motivated by the interest in using constraintbased technology in conjunction with ru ..."
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This article illustrates how constraint logic programming can be used to express data models in rulebased languages, including those based on graph patternmatching or unification to drive rule application. This is motivated by the interest in using constraintbased technology in conjunction with rulebased technology to provide a formally correct and effective—indeed, efficient!—operational base for the semantic web.
Combining Optimizations, Combining Theories
"... We consider the problem of how best to combine optimizations in imperative compilers. It is known that combined optimizations (or "superanalyses") can be strictly better than iterating separate improvement passes. We propose an explanation of why this is so by drawing connections between ..."
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We consider the problem of how best to combine optimizations in imperative compilers. It is known that combined optimizations (or "superanalyses") can be strictly better than iterating separate improvement passes. We propose an explanation of why this is so by drawing connections between program analysis and the algebraic and coalgebraic views of programs and processes. We argue that "optimistic" analyses decide coinductivelydefined relations and are based on bisimilarity.
EVERY COUNTABLE MODEL OF SET THEORY EMBEDS INTO ITS OWN CONSTRUCTIBLE UNIVERSE
"... Abstract. The main theorem of this article is that every countable model of set theory 〈M, ∈ M 〉, including every wellfounded model, is isomorphic to a submodel of its own constructible universe 〈L M, ∈ M 〉. In other words, there is an embedding j: M → L M that is elementary for quantifierfree a ..."
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Abstract. The main theorem of this article is that every countable model of set theory 〈M, ∈ M 〉, including every wellfounded model, is isomorphic to a submodel of its own constructible universe 〈L M, ∈ M 〉. In other words, there is an embedding j: M → L M that is elementary for quantifierfree assertions. It follows from the proof that the countable models of set theory are linearly preordered by embeddability: for any two countable models of set theory 〈M, ∈ M 〉 and 〈N, ∈ N 〉, either M is isomorphic to a submodel of N or conversely. Indeed, they are prewellordered by embeddability in ordertype exactly ω1+1. Specifically, the countable wellfounded models are ordered under embeddability exactly in accordance with the heights of their ordinals; every shorter model embeds into every taller model; every model of set theory M is universal for all countable wellfounded binary relations of rank at most Ord M; and every illfounded model of set theory is universal for all countable acyclic binary relations. Finally, strengthening a classical theorem of Ressayre, the same proof method shows that if M is any nonstandard model of PA, then every countable model of set theory—in particular, every model of ZFC plus large cardinals—is isomorphic to a submodel of the hereditarily finite sets 〈HF M, ∈ M 〉 of M. Indeed, 〈HF M, ∈ M 〉 is universal for all countable acyclic binary relations. 1.
Matter, Mind and Mechanics New models for dynamogenesis and rationality Synopsis of The Mechanical Foundations of Psychology and Economics
, 2000
"... The mathematical concepts of modern axiomatic rational mechanics apply more broadly than generally recognized, in that formal mechanical concepts apply directly and with slight adaptations to certain psychological and economic systems as well the familiar physical applications. These nonphysical app ..."
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The mathematical concepts of modern axiomatic rational mechanics apply more broadly than generally recognized, in that formal mechanical concepts apply directly and with slight adaptations to certain psychological and economic systems as well the familiar physical applications. These nonphysical applications provide new means for characterizing realistic notions of economic rationality and limits on reasoning abilities, translate psychology and economics into studies of new types of mechanical materials, and open traditional informal philosophical conceptions of materialism and dualism to new avenues of mathematical and experimental investigation. This semiformal article, which summarizes a detailed booklength formal treatment [13], briefly sketches modern mechanics, the adaptation of mechanical axioms to cover hybrid and discrete systems, an illustrative formalization of a representative rational psychological system from artificial intelligence, and the new perspective on psychology and economics,
unknown title
, 2006
"... www.elsevier.com/locate/tcs Processes as formal power series: A coinductive approach to denotational semantics � ..."
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www.elsevier.com/locate/tcs Processes as formal power series: A coinductive approach to denotational semantics �
C. Consistency and Incompleteness D. Stronger Axioms SET THEORY
"... Cardinal number Measure of the size of a set (possibly infinite). Choice axiom Principle that a set can be formed by making any number of arbitrary choices. Continuum hypothesis Statement that the set of real numbers has the smallest uncountable cardinality. Equinumerous Property of being in a onet ..."
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Cardinal number Measure of the size of a set (possibly infinite). Choice axiom Principle that a set can be formed by making any number of arbitrary choices. Continuum hypothesis Statement that the set of real numbers has the smallest uncountable cardinality. Equinumerous Property of being in a onetoone correspondence. Extensionality Principle that sets having the same members, however described, are identical. Ordinal number Measure of the length of a wellordering. Rank Ordinal number measuring the number of iterations of the powerset operation needed to construct a set. Wellordering Ordering for which any nonempty set has a least element. Set theory is that part of mathematics concerned with the abstract concepts of sets (i.e., collections of objects) and membership in sets. Thus the initial ideas from which set theory begins are extremely simple, and indeed are ideas that occur throughout mathematics. It is remarkable that from simple beginnings, a fascinating and useful theory emerges. And because set theory’s concepts occur throughout mathematics, it has been able to play a unifying role, providing both a common language and a foundational basis for mathematics. 1 I. THE ROLE OF SET THEORY IN MATHEMATICS