# Nonstandard analysis

alternative formulation of calculus using a logically rigorous notion of infinitesimal numbers

**Nonstandard analysis** is a logically precise method for the use of infinitesimals in calculus and mathematical analysis. Abraham Robinson originated the mathematics of nonstandard analysis in the 1960s.

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## Quotes

edit- There are three main tools in nonstandard analysis. One is the
*transference principle*, which roughly states that the same assertions of the formal language are true in the standard universe as in the nonstandard universe. It is typically used by proving a desired result in the nonstandard universe, and then, noting that the result is expressible in the language, concluding that it holds in the standard universe as well.

Another technique is*concurrence*. This is a logical technique that guarantees that the extended structure contains all possible completions, compactifications, and so forth.

The third technique is*internality*. A set*s*of elements of the nonstandard universe is*internal*if*s*itself is an element of the nonstandard universe; otherwiise,*s*is*external*. A surprislingly useful method of proof is one by reductio ad absurdum in which the contradiction is that some set one knows to be*external*would in fact be*internal*under the assumption being refuted.- Martin Davis:
*Applied Nonstandard Analysis*. 2014. p. 3. ISBN 9780486152349. (originally published in 1977)

- Martin Davis:

- Nonstandard analysis has proved to be a natural framework for studying the local properties of Banach spaces. The central construction in this approach is the nonstandard hull, introduced by Luxemburg ... . Not only is this a useful tool in studying the local theory of Banach space geometry, but also nonstandard hulls arise naturally in many other places within nonstandard analysis.
- C. Ward Henson and L. C. Moore Jr.: "Nonstandard analysis and the theory of Banach spaces".
*Nonstandard Analysis-Recent Developments*. Lecture Notes in Mathematics.**983**. 1983. pp. 27–112. doi: . ISBN 978-3-540-12279-1.

- C. Ward Henson and L. C. Moore Jr.: "Nonstandard analysis and the theory of Banach spaces".

- We describe and analyze a parametrization of fractal ‘‘curves’’ (i.e., fractal of topological dimension 1). The nondifferentiability of fractals and their infinite length forbid a complete description based on usual real numbers. We show that using nonstandard analysis it is possible to solve this problem: A class of nonstandard curves (whose standard part is the usual fractal) is defined so that a curvilinear coordinate along the fractal can be built, this being the first step towards the possible definition and study of a fractal space. We mention fields of physics to which such a formalism could be applied in the future.
- Laurent Nottale and J. Schneider: (1984). "Fractals and nonstandard analysis".
*Journal of Mathematical Physics***25**(5): 1296–1300. DOI:10.1063/1.526285.

- Laurent Nottale and J. Schneider: (1984). "Fractals and nonstandard analysis".

- It was in 1966 that A. Robinson's book ... on nonstandard analysis ... appeared. In it, a first rigorous foundation of the theory of infinitesmals was developed. In fact, A. Robinson had been using theorems in mathematical logic in the fifties to derive known mathematical results in a neoclassical way. His methods were based on the theory of models and in particular on the Löwenheim-Skolem theorem.
- Alain M. Robert:
*Nonstandard Analysis*. 2003. p. ix. ISBN 9780486432793. (French original published in 1985; English translation first published in 1988)

- Alain M. Robert:

- ... in the author's opinion this is the most important advantage of nonstandard analysis over standard analysis: To have convenient (almost "explicit") representations of certain objects like Hahn-Banach limits for which by standard methods more or less only their mere existence can be proved with the axiom of choice.
- Martin Andreas Väth:
*Nonstandard Analysis*. 2006. pp. 2–3. ISBN 9783764377748.

- Martin Andreas Väth: