Editing Well-order (section)

== Examples and counterexamples ==

=== Natural numbers ===
The standard ordering ≤ of the [[natural number]]s is a well ordering and has the additional property that every non-zero natural number has a unique predecessor.

Another well ordering of the natural numbers is given by defining that all even numbers are less than all odd numbers, and the usual ordering applies within the evens and the odds:

:<math>\begin{matrix}
0 & 2 & 4 & 6 & 8 & \dots & 1 & 3 & 5 & 7 & 9 & \dots
\end{matrix}</math>

This is a well-ordered set of [[order type]] {{math|''ω'' + ''ω''}}. Every element has a successor (there is no largest element).  Two elements lack a predecessor: 0 and 1.

=== 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}}.

=== Reals ===
The standard ordering ≤ of any [[Interval_(mathematics)|real interval]] is not a well ordering, since, for example, the [[open interval]] {{tmath|(0, 1) \subseteq [0, 1]}} does not contain a least element.   From the [[ZFC]] axioms of set theory (including the [[axiom of choice]]) one can show that there is a well order of the reals. Also [[Wacław Sierpiński]] proved that ZF + GCH (the [[generalized continuum hypothesis]]) imply the axiom of choice and hence a well order of the reals. Nonetheless, it is possible to show that the ZFC+GCH axioms alone are not sufficient to prove the existence of a definable (by a formula) well order of the reals.<ref>{{cite journal |author-link=Solomon Feferman |first=S. |last=Feferman |title=Some Applications of the Notions of Forcing and Generic Sets |journal=[[Fundamenta Mathematicae]] |volume=56 |issue=3 |year=1964 |pages=325–345 |doi=10.4064/fm-56-3-325-345 |url=https://eudml.org/doc/213821|doi-access=free }}</ref> However it is consistent with ZFC that a definable well ordering of the reals exists—for example, it is consistent with ZFC that [[V=L]], and it follows from ZFC+V=L that a particular formula well orders the reals, or indeed any set.

An uncountable subset of the real numbers with the standard ordering ≤ cannot be a well order: Suppose {{mvar|X}} is a subset of {{tmath|\R}} well ordered by {{char|≤}}. For each {{mvar|x}} in {{mvar|X}}, let {{math|''s''(''x'')}} be the successor of {{mvar|x}} in {{char|≤}} ordering on {{mvar|X}} (unless {{mvar|x}} is the last element of {{mvar|X}}). Let <math>A = \{(x,s(x)) \,|\, x \in X\}</math> whose elements are nonempty and disjoint intervals. Each such interval contains at least one rational number, so there is an [[injective function]] from {{mvar|A}} to {{tmath|\Q.}} There is an injection from {{mvar|X}} to {{mvar|A}} (except possibly for a last element of {{mvar|X}}, which could be mapped to zero later). And it is well known that there is an injection from {{tmath|\Q}} to the natural numbers (which could be chosen to avoid hitting zero). Thus there is an injection from {{mvar|X}} to the natural numbers, which means that {{mvar|X}} is countable. On the other hand, a countably infinite subset of the reals may or may not be a well order with the standard {{char|≤}}. For example,

* The natural numbers are a well order under the standard ordering {{char|≤}}.
* The set <math>\{1/n \,|\, n=1,2,3,\dots\}</math> has no least element and is therefore not a well order under standard ordering {{char|≤}}.

Examples of well orders:
*The set of numbers <math>\{-2^{-n} \,|\, 0 \leq n < \omega\}</math> has order type {{math|ω}}.
*The set of numbers <math>\{-2^{-n} - 2^{-m-n} \,|\, 0 \leq m,n < \omega \}</math> has order type {{math|''ω''<sup>2</sup>}}. The previous set is the set of [[limit point]]s within the set. Within the set of real numbers, either with the ordinary topology or the order topology, 0 is also a limit point of the set. It is also a limit point of the set of limit points.
*The set of numbers <math>\{-2^{-n} \,|\, 0 \leq n < \omega\} \cup \{1\}</math> has order type {{math|''ω'' + 1}}. With the [[order topology]] of this set, 1 is a limit point of the set, despite being separated from the only limit point 0 under the ordinary topology of the real numbers.