PlanetMath: hyperreal

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hyperreal

(Definition)

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

$\displaystyle \{a_n\} \sim \{b_n\} \iff \{n \in \mathbb{N}\mid a_n = b_n\} \in \mathcal{F}$

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

$\displaystyle \{a_n\} \leq \{b_n\} \iff \{n \in \mathbb{N}\mid a_n \leq b_n\} \in \mathcal{F}$

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

for all

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

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

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

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.

"hyperreal" is owned by djao.

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