well-order
التعريفات والمعاني
== English ==
=== Alternative forms ===
well order
=== Noun ===
well-order (plural well-orders)
(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,
X
_
{\displaystyle {\underline {X}}}
is well-order enriched iff every morphism set
X
_
(
X
,
Y
)
{\displaystyle {\underline {X}}(X,Y)}
carries a well-order
≤
X
Y
{\displaystyle \leq _{XY}}
such that
f
≨
X
Y
g
⇒
h
∙
f
≨
X
Y
h
∙
g
{\displaystyle f\lneqq _{XY}g\Rightarrow h\bullet f\lneqq _{XY}h\bullet g}
for every
h
:
Y
→
Z
{\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
X
{\displaystyle X}
is said to be a von Neumann well-order if for every
x
∈
X
{\displaystyle x\in X}
, we have
x
=
{
y
∈
X
|
y
<
x
}
{\displaystyle x=\{y\in X\vert y<x\}}
(that is
x
{\displaystyle x}
is equal to the set
P
r
e
d
(
x
)
{\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.
==== Synonyms ====
(type of total order): well-ordering
==== Hypernyms ====
(type of total order):
total order
partial order
preorder
==== Translations ====
=== Verb ===
well-order (third-person singular simple present well-orders, present participle well-ordering, simple past and past participle well-ordered)
(set theory, order theory, transitive) To impose a well-order on (a set).
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.
==== Translations ====
=== See also ===
ordinal number
tree
=== Further reading ===
Ordinal number on Wikipedia.Wikipedia
Well-ordering theorem on Wikipedia.Wikipedia
Well-ordering principle on Wikipedia.Wikipedia
Tree (set theory) on Wikipedia.Wikipedia
Well-founded relation on Wikipedia.Wikipedia