well-order

Noun

 * 1)  A total order of some set such that every nonempty subset contains a least element.
 * 2) * 1986, G. Richter, Noetherian semigroup rings with several objects, G. Karpilovsky (editor), Group and Semigroup Rings, Elsevier (North-Holland), page 237,
 * $$\underline{X}$$ is well-order enriched iff every morphism set $$\underline{X}(X,Y)$$ carries a well-order $$\leq_{XY}$$ such that
 * $$f\lneqq_{XY} g \Rightarrow h\bullet f\lneqq_{XY} h\bullet g$$
 * for every $$h : Y \rightarrow Z$$.
 * 1) * 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 (&empty;, &empty;) is a well-order.
 * 1) * 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 $$X$$ is said to be a von Neumann well-order if for every $$x\in X$$, we have $$x=\{y\in X\vert y< x\}$$ (that is $$x$$ is equal to the set $$\mathrm{Pred}(x)$$ consisting of its predecessors).
 * Clearly the examples listed by von Neumann above, namely
 * $$\empty,\quad \{\empty\},\quad \{\empty, \{\empty\}\},\quad \{\empty, \{\empty\}, \{\empty, \{\empty\}\}\},\quad\dots$$
 * are all von Neumann well-orders if ordered by the membership relation "$$\in$$," 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.

Translations

 * Danish: velordning
 * Dutch: welordening
 * Finnish: hyvinjärjestys
 * German: Wohlordnung
 * Hungarian: jólrendezés
 * Norwegian:
 * Bokmål: velordning
 * Nynorsk: velordning
 * Polish: porządek uporządkowany
 * Swedish: välordning

Verb

 * 1)  To impose a well-order on (a set).
 * 2) * 1950, Frederick Bagemihl (translator),, 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.
 * 1) * 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.
 * To carry the analogy a bit further, the axiom of choice implies the ability to well order any set.

Translations

 * Finnish: hyvinjärjestää
 * German: wohlordnen
 * Hungarian: jólrendez
 * Polish: uporządkowywać,