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Well-ordered Set
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order theory
   
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Roughly speaking, a well-ordered set is ordered in such a way that its elements can be considered one at a time, in order, and any time you haven't examined all of the elements, there's always a unique ''next'' element to consider.

Spelling note: The hyphen is frequently omitted in contemporary papers, yielding the spellings wellorder, '''wellordered''', '''wellordering'''.


EXAMPLES


  • The standard ordering ≤ of the Natural Number s is a well-ordering.

  • The standard ordering ≤ of the Integer s is not a well-ordering, since, for example, the set of Negative integers does not contain a least element.

  • The following relation ''R'' is a well-ordering of the integers: ''x R Y'' if and only if one of the following conditions holds:


:# ''x'' = 0
:# ''x'' is positive, and ''y'' is negative
:# ''x'' and ''y'' are both positive, and ''x'' ≤ ''y''
:# ''x'' and ''y'' are both negative, and ''y'' ≤ ''x''

R


0 1 2 3 4 ..... -1 -2 -3 .....

R