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table.yaml
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ID: INPUT{id.yaml}
Title: Hyperreal numbers
Definition: >
Given a free ultrafilter $U$ on the natural numbers $\mathbb{N}$,
the field of hyperreal numbers $^*\mathbb{R}$ CITE{WikiHyperreal}
can be defined as
the ultrapower $\mathbb{R}^\mathbb{N}/U$ CITE{WikiUltrafilter},
which is the set of all sequences of real numbers modulo
the equivalence relation $(a_n)_n \sim (b_n)_n$ if and only if
$\{n \mid a_n = b_n\} \in U$.
Parameters:
expression:
title: Name of constant
type: Symbolic
show-in-parameter-list: no
Comments:
comment-not-well-defined: >
Our current definition of $^*\mathbb{R}$ is not well-defined
as it depends on the choice of the free ultrafilter $U$.
This is against a basic principle of NumberDB according to which
every number in this database should be well-defined.
This could possibly be resolved using
the construction of Kanovei and Shelah CITE{KanShe04}.
comment-extension-of-R: >
$^*\mathbb{R}$ is an extension field of the real numbers $\mathbb{R}$:
An embedding $\mathbb{R} \to {}^*\mathbb{R}$ is given by
$r \mapsto (r)_n$.
comment-field: >
The hyperreal numbers become a field with respect to element-wise
operations.
(Except that for division, division by $0$ may happen at some indices $n$,
in which case once chooses an arbitrary real number as the result.)
comment-eps: >
The hyperreal number $\varepsilon = (1/n)_n$ is an infinitesimal:
$\varepsilon > 0$ and $\varepsilon < r$ for any positive real number $r$.
comment-1/eps: >
Similarly, $1/\varepsilon$ is an infinite hyperreal number:
$1/\varepsilon > r$ for any real number $r$.
Formulas:
Programs:
References:
KanShe04:
bib: >
Kanovei, Vladimir; Shelah, Saharon,
"A definable nonstandard model of the reals",
Journal of Symbolic Logic, 69: 159–164, (2004).
arXiv: math/0311165
doi: 10.2178/jsl/1080938834
Links:
WikiHyperreal:
title: "Wikipedia: Hyperreal number"
url: https://en.wikipedia.org/wiki/Hyperreal_number
WikiUltrafilter:
title: "Wikipedia: Ultraproduct"
url: https://en.wikipedia.org/wiki/Ultraproduct
Similar tables:
Keywords:
Tags:
- stub
- ring
- number system
- nonstandard analysis
- axiom of choice
- set theory
Data properties:
type: "*R"
Display properties:
Numbers:
1:
param-latex: $1$
number: >
(n: 1 for n in NN)
eps:
param-latex: $\varepsilon$
number: >
(n: 1/n for n in NN)
1/eps:
param-latex: $1/\varepsilon$
number: >
(n: n for n in NN)