Princeton, NJ: Princeton University Press, 1996. NonstandardĪnalysis for the Working Mathematician. "An Invitation to Nonstandard Analysis." In NonstandardĪnalysis and Its Applications (Ed. Keisler,Ĭalculus: An Infinitesimal Approach. Introduction to Nonstandard Real Analysis. Princeton, NJ: Princeton University Press, 1998. Robinson: The Creation of Nonstandard Analysis, A Personal and Mathematical Odyssey. Methods in Stochastic Analysis and Mathematical Physics. Examples include the use of nonstandard analysis in mathematical economics to describe the behavior of large economies and the use of nonstandard methods to give meaning to concepts that do not classically make sense, such as certain products of infinitely many independent, equally weighted random variables. In addition, in the nonstandard analysis community, there is a growing number of results that are not being translated into standard results, because the intuitive content of certain theorems is greater and/or clearer when left in nonstandard terminology. Crucially, however, the angle at which the nonstandard analyst looks at the axioms of analysis provides for an average case reduction in complexity that provides shorter proofs of various results, and will one day lead to the proof of a result which is not accessible to classical mathematics without nonstandard methods, precisely because its classical proof is too long to write down in the length of time humans will reside on Earth. Loosely, nonstandard methods replace higher-order concepts with first-order analogs. One may refer to the measure as an element, for it is in fact an element of. In the first-order theory of this superstructure, Sets, with the only fundamental relation being the membership relation, then Texts on nonstandard analysis, as the union of (roughly) a tower of iterated power Way that uses higher-order logic, and the measure is a sort of "higher order object", since it isīut if one forms the superstructure that has as its individuals the members of and the real numbers, and is constructed as described in typical This way of looking at measure spaces is a A measure space consists of a set, together with a measure, which is a function from some sigma-algebra of subsets of into the reals. In classical analysis, one studies measure The axioms used in nonstandard analysis are first-order set theoretical axioms, but many of the topics studied in classical analysis that are axiomatized with higher-orderĪxioms can be reformulated in set theoretical terms in a first-order axiomatization.Īs an example, consider the notion of a measure on a Has been applied in areas such as Banach spaces,ĭifferential equations, probability theory, mathematical economics, and mathematical The theory has since been investigated for its own sake and Abraham Robinson developed nonstandardĪnalysis in the 1960s. The existence of "genuine infinitesimals," which are numbers that are less Nonstandard analysis is a branch of mathematical logic which introduces hyperreal numbers to allow for
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