Editing Well-order (section)

=== Integers ===
Unlike the standard ordering ≤ of the [[natural number]]s, the standard ordering ≤ of the [[integer]]s is not a well ordering, since, for example, the set of [[negative number|negative]] integers does not contain a least element.

The following [[binary relation]] {{mvar|R}} is an example of well ordering of the integers: {{mvar|x R y}} [[if and only if]] one of the following conditions holds:
# {{math|1=''x'' = 0}}
# {{mvar|x}} is positive, and {{mvar|y}} is negative
# {{mvar|x}} and {{mvar|y}} are both positive, and {{math|''x'' ≤ ''y''}}
# {{mvar|x}} and {{mvar|y}} are both negative, and {{math|{{abs|''x''}} ≤ {{abs|''y''}}}}
This relation {{mvar|R}} can be visualized as follows: 
:<math>\begin{matrix}
0 & 1 & 2 & 3 & 4 & \dots & -1 & -2 & -3 & \dots
\end{matrix}</math>
{{mvar|R}} is isomorphic to the [[ordinal number]] {{math|''ω'' + ''ω''}}.

Another relation for well ordering the integers is the following definition: <math>x \leq_z y</math> [[if and only if]] 
:<math>|x| < |y| \qquad \text{or}  \qquad |x| = |y| \text{ and } x \leq y.</math>
This well order can be visualized as follows: 
:<math>\begin{matrix}
0 & -1 & 1 & -2 & 2 & -3 & 3 & -4 & 4 & \dots
\end{matrix}</math>

This has the [[order type]] {{mvar|ω}}.