Editing Nonstandard analysis (section)

== Introduction ==
A non-zero element of an [[ordered field]] <math>\mathbb F</math> is infinitesimal if and only if its [[absolute value]] is smaller than any element of <math>\mathbb F</math> that is of the form <math>\frac{1}{n}</math>, for <math>n</math> a standard [[natural number]]. Ordered fields that have infinitesimal elements are also called [[non-Archimedean ordered field|non-Archimedean]]. More generally, nonstandard [[mathematical analysis|analysis]] is any form of mathematics that relies on [[nonstandard model]]s and the [[transfer principle]]. A field that satisfies the transfer principle for real numbers is called a [[real closed field]], and nonstandard [[real analysis]] uses these fields as ''nonstandard models'' of the real numbers.

Robinson's original approach was based on these nonstandard models of the field of real numbers. His classic foundational book on the subject ''Nonstandard Analysis'' was published in 1966 and is still in print.<ref name="NSA2">{{cite book| last=Robinson |first=Abraham|title=Nonstandard analysis|year=1996|edition=Revised|publisher=Princeton University Press|isbn=0-691-04490-2}}</ref> On page 88, Robinson writes:

<blockquote>The existence of nonstandard models of arithmetic was discovered by [[Thoralf Skolem]] (1934). Skolem's method foreshadows the [[ultrapower]] construction [...]</blockquote>

Several technical issues must be addressed to develop a calculus of infinitesimals. For example, it is not enough to construct an ordered field with infinitesimals. See the article [[Hyperreal number]] for a discussion of some of the relevant ideas.