# hyperreal

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

Fix once and for all a nonprincipal ultrafilter $\mathcal{F}$ on the set $\mathbb{N}$ of natural numbers. Let $\sim$ be the equivalence relation on the set $\mathbb{R}^{\mathbb{N}}$ of sequences of real numbers given by

 $\{a_{n}\}\sim\{b_{n}\}\iff\{n\in\mathbb{N}\mid a_{n}=b_{n}\}\in\mathcal{F}$

Let ${}^{*}\mathbb{R}$ be the set of equivalence classes of $\mathbb{R}^{\mathbb{N}}$ under the equivalence relation $\sim$. The set ${}^{*}\mathbb{R}$ is called the set of hyperreals. It is a field under coordinatewise addition and multiplication:

 $\displaystyle\{a_{n}\}+\{b_{n}\}$ $\displaystyle=$ $\displaystyle\{a_{n}+b_{n}\}$ $\displaystyle\{a_{n}\}\cdot\{b_{n}\}$ $\displaystyle=$ $\displaystyle\{a_{n}\cdot b_{n}\}$

The field ${}^{*}\mathbb{R}$ is an ordered field under the ordering relation

 $\{a_{n}\}\leq\{b_{n}\}\iff\{n\in\mathbb{N}\mid a_{n}\leq b_{n}\}\in\mathcal{F}$

The real numbers embed into ${}^{*}\mathbb{R}$ by the map sending the real number $x\in\mathbb{R}$ 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 $\mathbb{R}$ as a subset of ${}^{*}\mathbb{R}$ under this embedding.

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

• limited if $a for some real numbers $a,b\in\mathbb{R}$

• positive unlimited if $x>a$ for all real numbers $a\in\mathbb{R}$

• negative unlimited if $x for all real numbers $a\in\mathbb{R}$

• unlimited if it is either positive unlimited or negative unlimited

• positive infinitesimal if $0 for all positive real numbers $a\in\mathbb{R}^{+}$

• negative infinitesimal if $a for all negative real numbers $a\in\mathbb{R}^{-}$

• infinitesimal if it is either positive infinitesimal or negative infinitesimal

For any subset $A$ of $\mathbb{R}$, the set ${}^{*}A$ is defined to be the subset of ${}^{*}\mathbb{R}$ consisting of equivalence classes of sequences $\{a_{n}\}$ such that

 $\{n\in\mathbb{N}\mid a_{n}\in A\}\in\mathcal{F}.$

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.

 Title hyperreal Canonical name Hyperreal Date of creation 2013-03-22 12:35:45 Last modified on 2013-03-22 12:35:45 Owner djao (24) Last modified by djao (24) Numerical id 4 Author djao (24) Entry type Definition Classification msc 26E35 Synonym nonstandard real Synonym non-standard real Related topic Infinitesimal2 Defines nonprincipal ultrafilter Defines infinitesimal Defines hypernatural Defines hyperinteger Defines hyperrational Defines hyperfinite