Editing Nonstandard analysis (section)

== Basic definitions ==
In this section we outline one of the simplest approaches to defining a hyperreal field <math>^*\mathbb{R}</math>. Let <math>\mathbb{R}</math> be the field of real numbers, and let <math>\mathbb{N}</math> be the [[semiring]] of natural numbers. Denote by <math>\mathbb{R}^{\mathbb{N}}</math> the set of sequences of real numbers. A field <math>^*\mathbb{R}</math> is defined as a suitable quotient of <math>\mathbb{R}^\mathbb{N}</math>, as follows. Take a nonprincipal [[ultrafilter]] <math>F \subseteq P(\mathbb{N})</math>. In particular, <math>F</math> contains the [[Fréchet filter]]. Consider a pair of sequences

:<math>u = (u_n), v = (v_n) \in \mathbb{R}^\mathbb{N}</math>

We say that <math>u</math> and <math>v</math> are equivalent if they coincide on a set of indices that is a member of the ultrafilter, or in formulas:

:<math>\{n \in \mathbb{N} : u_n = v_n\} \in F</math>

The quotient of <math>\mathbb{R}^\mathbb{N}</math> by the resulting equivalence relation is a hyperreal field <math>^*\mathbb{R}</math>, a situation summarized by the formula <math>^*\mathbb{R} = {\mathbb{R}^\mathbb{N}}/{F}</math>.