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Integers

Unlike the standard ordering ≤ of the natural numbers, the standard ordering ≤ of the integers is not a well ordering, since, for example, the set of negative integers does not contain a least element.

The following binary relation R is an example of well ordering of the integers: x R y if and only if one of the following conditions holds:

This relation R can be visualized as follows:

R is isomorphic to the ordinal number ω + ω.

Another relation for well ordering the integers is the following definition: x ≤ z y {\displaystyle x\leq _{z}y} if and only if

This well order can be visualized as follows:

This has the order type ω.