Hyperfinite set

<h2 id="ultrapower-construction">Ultrapower construction</h2>
<p>In terms of the <a href="/facts/Ultrapower/d4cT10hd">ultrapower</a> construction, the hyperreal line *R is defined as the collection of <a href="/facts/Equivalence_class/1d2sh0TB">equivalence classes</a> of sequences 
  
    
      
        ⟨
        
          u
          
            n
          
        
        ,
        n
        =
        1
        ,
        2
        ,
        …
        ⟩
      
    
    {\displaystyle \langle u_{n},n=1,2,\ldots \rangle }
  
 of real numbers <i>u</i><i>n</i>.  Namely, the equivalence class defines a hyperreal, denoted 
  
    
      
        [
        
          u
          
            n
          
        
        ]
      
    
    {\displaystyle [u_{n}]}
  
 in Goldblatt's notation.  Similarly, an arbitrary hyperfinite set in *R is of the form 
  
    
      
        [
        
          A
          
            n
          
        
        ]
      
    
    {\displaystyle [A_{n}]}
  
, and is defined by a sequence 
  
    
      
        ⟨
        
          A
          
            n
          
        
        ⟩
      
    
    {\displaystyle \langle A_{n}\rangle }
  
 of finite sets 
  
    
      
        
          A
          
            n
          
        
        ⊆
        
          R
        
        ,
        n
        =
        1
        ,
        2
        ,
        …
      
    
    {\displaystyle A_{n}\subseteq \mathbb {R} ,n=1,2,\ldots }
  
<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a>
</p>

<h2 id="external-links">External links</h2>
<ul><li>M. Insall. <a href="https://mathworld.wolfram.com/HyperfiniteSet.html">"Hyperfinite Set"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>J. E. Rubio (1994). Optimization and nonstandard analysis. Marcel Dekker. p. 110. ISBN 0-8247-9281-5. <a href="0-8247-9281-5" target="_blank">0-8247-9281-5</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>R. Chuaqui (1991). Truth, possibility, and probability: new logical foundations of probability and statistical inference. Elsevier. pp. 182–3. ISBN 0-444-88840-3. <a href="0-444-88840-3" target="_blank">0-444-88840-3</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>R. Chuaqui (1991). Truth, possibility, and probability: new logical foundations of probability and statistical inference. Elsevier. pp. 182–3. ISBN 0-444-88840-3. <a href="0-444-88840-3" target="_blank">0-444-88840-3</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>R. Chuaqui (1991). Truth, possibility, and probability: new logical foundations of probability and statistical inference. Elsevier. pp. 182–3. ISBN 0-444-88840-3. <a href="0-444-88840-3" target="_blank">0-444-88840-3</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>L. Ambrosio; et al. (2000). Calculus of variations and partial differential equations: topics on geometrical evolution problems and degree theory. Springer. p. 203. ISBN 3-540-64803-8. <a href="3-540-64803-8" target="_blank">3-540-64803-8</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Rob Goldblatt (1998). Lectures on the hyperreals. An introduction to nonstandard analysis. Springer. p. 188. ISBN 0-387-98464-X. <a href="0-387-98464-X" target="_blank">0-387-98464-X</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
</ol>

Hyperfinite set open-in-new

Hyperfinite set