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