18,068 research outputs found

    Pincherle's theorem in Reverse Mathematics and computability theory

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    We study the logical and computational properties of basic theorems of uncountable mathematics, in particular Pincherle's theorem, published in 1882. This theorem states that a locally bounded function is bounded on certain domains, i.e. one of the first 'local-to-global' principles. It is well-known that such principles in analysis are intimately connected to (open-cover) compactness, but we nonetheless exhibit fundamental differences between compactness and Pincherle's theorem. For instance, the main question of Reverse Mathematics, namely which set existence axioms are necessary to prove Pincherle's theorem, does not have an unique or unambiguous answer, in contrast to compactness. We establish similar differences for the computational properties of compactness and Pincherle's theorem. We establish the same differences for other local-to-global principles, even going back to Weierstrass. We also greatly sharpen the known computational power of compactness, for the most shared with Pincherle's theorem however. Finally, countable choice plays an important role in the previous, we therefore study this axiom together with the intimately related Lindel\"of lemma.Comment: 43 pages, one appendix, to appear in Annals of Pure and Applied Logi

    Revealed Preference with Stochastic Demand Correspondence

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    We unify and expand the theory of consumer’s behavior, based on Samuelson’s Weak Axiom of Revealed Preference, to permit simultaneously both random choice and non-singleton choice sets. We provide a consistency postulate for demand behavior when such behavior is represented in terms of a stochastic demand correspondence. When the consumer spends her entire wealth, our rationality postulate is equivalent to a condition we term stochastic substitutability. This equivalence generates: (i) Samuelson’s Substitution Theorem, (ii) the central result in Bandyopadhyay, Dasgupta and Pattanaik (2004) and (iii) a version pertinent to deterministic demand correspondences (which independently yields Samuelson’s Substitution Theorem), as alternative special cases. Relevant versions of the non-positivity of the own substitution effect, the demand theorem and homogeneity of degree zero in prices and wealth for the consumer’s demand behavior, also follow as corollaries in every case.Stochastic demand correspondence, weak axiom of revealed preference, weak axiom of stochastic revealed preference, general substitution theorem, demand theorem.

    Reverse Mathematics and parameter-free Transfer

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    Recently, conservative extensions of Peano and Heyting arithmetic in the spirit of Nelson's axiomatic approach to Nonstandard Analysis, have been proposed. In this paper, we study the Transfer axiom of Nonstandard Analysis restricted to formulas without parameters. Based on this axiom, we formulate a base theory for the Reverse Mathematics of Nonstandard Analysis and prove some natural reversals, and show that most of these equivalences do not hold in the absence of parameter-free Transfer.Comment: 22 pages; to appear in Annals of Pure and Applied Logi
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