Nonstandard analysis has been used recently in major results,such as Jin’s sumset theorem in additive combinatoricsand Breuillard–Green–Tao’s work on the structure of approximate groups. However, its roots go back to Robinson’s formalization of the infinitesimal approach to calculus. After first illustrating its very basic uses in calculus in “Calculus with Infinitesimals,” we go on to highlight a selection of its more serious achievements in “Selected Classical and Recent Applications,” including the aforementioned work of Jin and Breuillard–Green–Tao. After presenting a simple axiomatic approach to nonstandard analysis in “Axioms for Nonstandard Extensions” we examine Jin’s theorem in more detail in “The Axioms in Action: Jin’s Theorem.” Finally, in “The Ultraproduct Construction” we discuss how these axioms can be justified with a particular concrete construction (akin to the verification of the axioms for the real field using Dedekind cuts or Cauchy sequences), and in “Other Approaches” we compare our axiomatic approach to other approaches. While brief, our hope is that this survey can quickly give the reader a sense of both how nonstandard methods are being used today and how these methods can be rigorously presented and justified.
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