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Foundations Of Nonstandard Analysis  A Gentle Introduction to Nonstandard Extemsions
 In Nonstandard analysis (Edinburgh
"... this paper is to describe the essential features of the resulting frameworks without getting bogged down in technicalities of formal logic and without becoming dependent on an explicit construction of a specific field ..."
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this paper is to describe the essential features of the resulting frameworks without getting bogged down in technicalities of formal logic and without becoming dependent on an explicit construction of a specific field
On the Foundations of Nonstandard Mathematics
"... In this paper we survey various settheoretic approaches that have been proposed over the last thirty years as foundational frameworks for the use of nonstandard methods in mathematics. Introduction. Since the early developments of calculus, infinitely small and infinitely large numbers have been th ..."
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In this paper we survey various settheoretic approaches that have been proposed over the last thirty years as foundational frameworks for the use of nonstandard methods in mathematics. Introduction. Since the early developments of calculus, infinitely small and infinitely large numbers have been the object of constant interest and great controversy in the history of mathematics. In fact, while on the one hand fundamental results in the differential and integral calculus were first obtained by reasoning informally
ISSN: 09316558A REPRESENTATIVE INDIVIDUAL FROM ARROVIAN AGGREGATION OF PARAMETRIC INDIVIDUAL UTILITIES
, 2009
"... ABSTRACT. This article investigates the representativeagent hypothesis for a population which faces a collective choice from a given finitedimensional space of alternatives. Each individual’s preference ordering is assumed to admit a utility representaion through an element of an exogenously given ..."
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ABSTRACT. This article investigates the representativeagent hypothesis for a population which faces a collective choice from a given finitedimensional space of alternatives. Each individual’s preference ordering is assumed to admit a utility representaion through an element of an exogenously given set of admissible utility functions. In addition, we assume that (i) the class of admissible utility functions allows for a smooth parametrization and only consists of strictly concave functions, (ii) the population is infinite, and (iii) the social welfare function satisfies Arrovian rationality axioms. We prove that there exists an admissible utility function r, called representative utility function, such that any alternative which maximizes r also maximizes the social welfare function. Given the structural similarities among the admissible utility functions (due to parametrization), we argue that the representative utility function can be interpreted as belonging to an — actual or invisible — individual. The existence proof for the representative utility function utilizes a special nonstandard model of the reals, viz. the ultrapower of the reals with respect to the ultrafilter of decisive coalitions; this construction explicitly determines the parameter vector of the representative utility function.