Well-ordering principle

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the well-ordering principle states that every non-empty subset of nonnegative integers contains a <a href="/facts/Least_element/JVn2B3mo">least element</a>. In other words, the set of nonnegative integers is <a href="/facts/Well-order/GrWEBWlW">well-ordered</a> by its "natural" or "magnitude" order in which 
  
    
      
        x
      
    
    {\displaystyle x}
  
 precedes 
  
    
      
        y
      
    
    {\displaystyle y}
  
 if and only if 
  
    
      
        y
      
    
    {\displaystyle y}
  
 is either 
  
    
      
        x
      
    
    {\displaystyle x}
  
 or the sum of 
  
    
      
        x
      
    
    {\displaystyle x}
  
 and some nonnegative integer (other orderings include the ordering 
  
    
      
        2
        ,
        4
        ,
        6
        ,
        .
        .
        .
      
    
    {\displaystyle 2,4,6,...}
  
; and 
  
    
      
        1
        ,
        3
        ,
        5
        ,
        .
        .
        .
      
    
    {\displaystyle 1,3,5,...}
  
).
</p><p>The phrase "well-ordering principle" is sometimes taken to be synonymous with the "<a href="/facts/Well-ordering_theorem/teE1cYkm">well-ordering theorem</a>". On other occasions it is understood to be the proposition that the set of <a href="/facts/Integers/PUwwrawx">integers</a> 
  
    
      
        {
        …
        ,
        −
        2
        ,
        −
        1
        ,
        0
        ,
        1
        ,
        2
        ,
        3
        ,
        …
        }
      
    
    {\displaystyle \{\ldots ,-2,-1,0,1,2,3,\ldots \}}
  
 contains a <a href="/facts/Well-order/GrWEBWlW">well-ordered</a> subset, called the <a href="/facts/Natural_numbers/u65og8JE">natural numbers</a>, in which every nonempty subset contains a least element.
</p>

Well-ordering principle open-in-new

Well-ordering principle