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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 $r$.

comment-schroedingers-infinity: >
    The hyperreal $S = ((-1)^n n)_n$ is either $1/\varepsilon$ or its negative,
    depending on the choice of $U$.
    We might call $S$ "Schrödinger's infinity".
    
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)
  S:
    param-latex: $S$
    number: >
      (n: (-1)^n * n for n in NN)

— preview —

Hyperreal numbers

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Numbers

$1$:

(n: 1 for n in NN)

$\varepsilon$:

(n: 1/n for n in NN)

$1/\varepsilon$:

(n: n for n in NN)

$S$:

(n: (-1)^n * n for n in NN)

Definition

Given a free ultrafilter $U$ on the natural numbers $\mathbb{N}$, the field of hyperreal numbers $^*\mathbb{R}$ [2] can be defined as the ultrapower $\mathbb{R}^\mathbb{N}/U$ [3], 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 $\{n \mid a_n = b_n\} \in U$.

Comments

(1)

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 [1].

(2)

$^*\mathbb{R}$ is an extension field of the real numbers $\mathbb{R}$: An embedding $\mathbb{R} \to {}^*\mathbb{R}$ is given by $r \mapsto (r)_n$.

(3)

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.)

(4)

The hyperreal number $\varepsilon = (1/n)_n$ is an infinitesimal: $\varepsilon > 0$ and $\varepsilon < r$ for any positive real number $r$.

(5)

Similarly, $1/\varepsilon$ is an infinite hyperreal number: $1/\varepsilon > r$ for any real number $r$.

(6)

The hyperreal $S = ((-1)^n n)_n$ is either $1/\varepsilon$ or its negative, depending on the choice of $U$. We might call $S$ "Schrödinger's infinity".

References

[1]

Kanovei, Vladimir; Shelah, Saharon, "A definable nonstandard model of the reals", Journal of Symbolic Logic, 69: 159–164, (2004). (arXiv) (doi)

Links

[2]

Wikipedia: Hyperreal number

[3]

Wikipedia: Ultraproduct

Data properties

Entries are of type: hyperreal number