Nonstandard analysis is a branch of mathematical logic which introduces hyperreal numbers to allow for the existence of "genuine infinitesimals," which are numbers that are less than 1/2, 1/3, 1/4, 1/5, ..., but greater than 0. Abraham Robinson developed nonstandard analysis in the 1960s. The theory has since been investigated for its own sake and has been applied in areas such as Banach spaces, differential equations, probability theory, mathematical economics, and mathematical physics.
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-order
axioms can be reformulated in set theoretical terms in a first-order axiomatization.
As an example, consider the notion of a measure on a
set. In classical analysis, one studies measure
spaces. 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. This way of looking at measure spaces is a
way that uses higher-order logic, and the measure 
is a sort of "higher order object", since it is
not an element of 
.
But if one forms the superstructure 
that has as its individuals the members of 
and the real numbers, and is constructed as described in typical
texts on nonstandard analysis, as the union of (roughly) a tower of iterated power
sets, with the only fundamental relation being the membership relation, then
in the first-order theory of this superstructure,
one may refer to the measure 
as an element, for it is in fact an element of 
.
Loosely, nonstandard methods replace higher-order concepts with first-order analogs. It looks at them from a different angle. 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.
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. 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.
