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:

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