Book Chapter10.1007/978-94-009-3895-3_15
Choice Processes, Computability and Complexity: Computable Choice Functions
Hans W. Gottinger
- 01 Jan 1988
- pp 287-300
TL;DR: In this paper, the authors model human choice processes by computational procedures and by representations of computational theory, to the extent that "human rationality" and "human problem-solving" has been taken as an anchor point for constructing artificial intelligence.
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Abstract: Ever since choice theory has established itself as part of economic theory and mathematical economics there have been attempts to axiozatize it on the basis of set theory and topology. To the extent that “human rationality” and “human problem-solving” has been taken as an anchor point for constructing “artificial intelligence” it would be natural to model human choice processes by computational procedures and by representations of computational theory.
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References
•Book
Computation: Finite and Infinite Machines
Marvin Minsky
- 01 Jan 1967
TL;DR: In this article, the authors present an abstract theory that categorically and systematically describes what all these machines can do and what they cannot do, giving sound theoretical or practical grounds for each judgment, and the abstract theory tells us in no uncertain terms that the machines' potential range is enormous and that its theoretical limitations are of the subtlest and most elusive sort.
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Institute for Mathematical Studies in the Social Sciences
Robert L. Smith,Lee H. Blaine +1 more
- 01 Jan 1975
TL;DR: ExCHECK is a system for developing mathematically-based CAl courses that is currently being used at Stanford University to teach a college-credit course in axiomatic set theory and describes the system's instructional uses.
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Lacombe Daniel. Quelques procédés de définition en topologie récursive. Constructivity in mathematics, Proceedings of the colloquium held at Amsterdam, 1957, edited by Heyting A., Studies in logic and the foundations of mathematics, North-Holland Publishing Company, Amsterdam 1959, pp. 129–158.
Abstract: In the beginning the author sketches a proof of the equivalence of the set theory of Bernays I I I 49, where two membership relations e and r\ are used, to that of Godel VI 112 and Mostowski IV 129, where only e is used. The author proves, in a slightly different but hardly more illuminating way, the results of Rosser-Wang XVI 145, Novak XVI 273, and Mostowski XVI 274, which establish the essential equivalence of the Zermelo-Fraenkel set theory to the von Neumann-Bernays set theory. Later the author defines, within the framework of the von Neumann-Bernays set theory, the notion of a class being nameable, or definable, relative to sets. He mentions that it is consistent to assume that all classes are nameable relative to sets, and from this, by a simple diagonal argument, he shows that there is a sentence of the form 3BVy(y « B <-• 3A(6(A, y))), where 0(A, y) contains no class variables other than the free variable A, which is not a theorem of the von Neumann-Bernays set theory (this is a new proof of a result somewhat weaker than that of Mostowski XVI 274). Each class nameable relative to sets can be given a name which is a set. The relation 'x is a member of the class whose name is y' is expressible by a formula of the form 3A6(A, x, y), where 0(A, x, y) contains no class variable other than the free variable A, but is shown by the author not to be expressible by a formula without class variables. The author also mentions some consequences of the result of Myhill XX 80 that the hypothesis that all classes are nameable is consistent with the von Neumann-Bernays set theory. I t is known that the proof of Myhill XX 80 establishes his result only upon the assumption that the von Neumann-Bernays set theory stays consistent when it is augmented by Godel's axiom of constructibility, V = L, and by the axiom schema of induction 6(0) A Vn(6(n) ->• 6(n 41)) -»Vn0(n), where 6 is any formula (which may contain also bound class variables). AZRIEL LEVY
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