An ultrafilter $\F$ on a set $I$ is called nonprincipal if no finite subsets of $I$ are in $\F$

Fix once and for all a nonprincipal ultrafilter $\F$ on the set $\N$ of natural numbers. Let $\sim$ be the equivalence relation on the set $\R^\N$ of sequences of real numbers given by $$ \{a_n\} \sim \{b_n\} \iff \{n \in \N \mid a_n = b_n\} \in \F $$ Let $^*\R$ be the set of equivalence classes of $\R^\N$ under the equivalence relation $\sim$ The set $^*\R$ is called the set of hyperreals. It is a field under coordinatewise addition and multiplication: \begin{eqnarray*} \{a_n\} + \{b_n\} & = & \{a_n+b_n\} \\ \{a_n\} \cdot \{b_n\} & = & \{a_n\cdot b_n\} \end{eqnarray*}The field $^*\R$ is an ordered field under the ordering relation $$ \{a_n\} \leq \{b_n\} \iff \{n \in \N \mid a_n \leq b_n\} \in \F $$ The real numbers embed into $^*\R$ by the map sending the real number $x \in \R$ to the equivalence class of the constant sequence given by $x_n := x$ for all $n$ In what follows, we adopt the convention of treating $\R$ as a subset of $^*\R$ under this embedding.

A hyperreal $x \in\,^*\R$ is:

• limited if $a < x < b$ for some real numbers $a,b \in \R$
• positive unlimited if $x > a$ for all real numbers $a \in \R$
• negative unlimited if $x < a$ for all real numbers $a \in \R$
• unlimited if it is either positive unlimited or negative unlimited
• positive infinitesimal if $0 < x < a$ for all positive real numbers $a \in \R^+$
• negative infinitesimal if $a < x < 0$ for all negative real numbers $a \in \R^-$
• infinitesimal if it is either positive infinitesimal or negative infinitesimal

For any subset $A$ of $\R$ the set $^*A$ is defined to be the subset of $^*\R$ consisting of equivalence classes of sequences $\{a_n\}$ such that $$ \{n \in \N \mid a_n \in A\} \in \F. $$ The sets $^*\mathbb{N}$ $^*\mathbb{Z}$ and $^*\mathbb{Q}$ are called hypernaturals, hyperintegers, and hyperrationals, respectively. An element of $^*\mathbb{N}$ is also sometimes called hyperfinite.