# well-order

## English

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### Noun

well-order (plural well-orders)

1. (set theory, order theory) A total order of some set such that every nonempty subset contains a least element.
• 1986, G. Richter, Noetherian semigroup rings with several objects, G. Karpilovsky (editor), Group and Semigroup Rings, Elsevier (North-Holland), page 237,
${\displaystyle {\underline {X}}}$ is well-order enriched iff every morphism set ${\displaystyle {\underline {X}}(X,Y)}$ carries a well-order ${\displaystyle \leq _{XY}}$ such that
${\displaystyle f\lneqq _{XY}g\Rightarrow h\bullet f\lneqq _{XY}h\bullet g}$
for every ${\displaystyle h:Y\rightarrow Z}$.
• 2001, Robert L. Vaught, Set Theory: An Introduction, Springer (Birkhäuser), 2nd Edition, Softcover, page 71,
Some simple facts and terminology about well-orders were already given in and just before 1.8.4. Here are some more: In a well-order A, every element x is clearly of just one of these three kinds: x is the first element; x is a successor element - i.e., x has an immediate predecessor; or x is a limit element - i.e., x has a predecessor but no immediate predecessor. The structure (∅, ∅) is a well-order.
• 2014, Abhijit Dasgupta, Set Theory: With an Introduction to Real Point Sets, Springer (Birkhäuser), page 378,
Definition 1226 (Von Neumann Well-Orders). A well-order ${\displaystyle X}$ is said to be a von Neumann well-order if for every ${\displaystyle x\in X}$, we have ${\displaystyle x=\{y\in X\vert y (that is ${\displaystyle x}$ is equal to the set ${\displaystyle \mathrm {Pred} (x)}$ consisting of its predecessors).
Clearly the examples listed by von Neumann above, namely
${\displaystyle \emptyset ,\quad \{\emptyset \},\quad \{\emptyset ,\{\emptyset \}\},\quad \{\emptyset ,\{\emptyset \},\{\emptyset ,\{\emptyset \}\}\},\quad \dots }$
are all von Neumann well-orders if ordered by the membership relation "${\displaystyle \in }$," and the process can be iterated through the transfinite. Our immediate goal is to show that these and only these are the von Neumann well-orders, with exactly one von Neumann well-order for each ordinal (order type of a well-order). This is called the existence and uniqueness result for the von Neumann well-orders.

### Verb

well-order (third-person singular simple present well-orders, present participle well-ordering, simple past and past participle well-ordered)

1. (set theory, order theory, transitive) To impose a well-order on (a set).
The set of positive integers is well-ordered by the relation ≤.
• 1950, Frederick Bagemihl (translator), Erich Kamke, Theory of Sets, 2006, Dover (Dover Phoenix), page 111,
Starting from these special well-ordered subsets, it is then possible to well-order the entire set.
• 1975 [The Williams & Wilkins Company], Dennis Sentilles, A Bridge to Advanced Mathematics, Dover, 2011, page 182,
To carry the analogy a bit further, the axiom of choice implies the ability to well order any set.
• 2006, Charalambos D. Aliprantis, Kim C. Border, Infinite Dimensional Analysis: A Hitchhiker's Guide, Springer, 3rd Edition, page 18,
Then ${\displaystyle \leq _{C}}$ is a well defined order on ${\displaystyle C}$, and ${\displaystyle (C,\leq _{C})}$ belongs to ${\displaystyle {\mathcal {X}}}$ (that is, ${\displaystyle \leq _{C}}$ well orders ${\displaystyle C}$) and is an upper bound for ${\displaystyle {\mathcal {C}}}$.