Well-order
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a well-order (or well-ordering or well-order relation) on a set is a total ordering on with the property that every non-empty
subset In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...
of has a least element in this ordering. The set together with the ordering is then called a well-ordered set (or woset). In some academic articles and textbooks these terms are instead written as wellorder, wellordered, and wellordering or well order, well ordered, and well ordering. Every non-empty well-ordered set has a least element. Every element of a well-ordered set, except a possible
greatest element In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined duality (order theory), dually ...
, has a unique successor (next element), namely the least element of the subset of all elements greater than . There may be elements, besides the least element, that have no predecessor (see below for an example). A well-ordered set contains for every subset with an
upper bound In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is every element of . Dually, a lower bound or minorant of is defined to be an element of that is less ...
a least upper bound, namely the least element of the subset of all upper bounds of in . If ≤ is a non-strict well ordering, then < is a strict well ordering. A relation is a strict well ordering if and only if it is a well-founded strict total order. The distinction between strict and non-strict well orders is often ignored since they are easily interconvertible. Every well-ordered set is uniquely order isomorphic to a unique
ordinal number In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the leas ...
, called the order type of the well-ordered set. The well-ordering theorem, which is equivalent to the axiom of choice, states that every set can be well ordered. If a set is well ordered (or even if it merely admits a well-founded relation), the proof technique of transfinite induction can be used to prove that a given statement is true for all elements of the set. The observation that the natural numbers are well ordered by the usual less-than relation is commonly called the well-ordering principle (for natural numbers).


Ordinal numbers

Every well-ordered set is uniquely order isomorphic to a unique
ordinal number In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the leas ...
, called the order type of the well-ordered set. The position of each element within the ordered set is also given by an ordinal number. In the case of a finite set, the basic operation of counting, to find the ordinal number of a particular object, or to find the object with a particular ordinal number, corresponds to assigning ordinal numbers one by one to the objects. The size (number of elements,
cardinal number In mathematics, a cardinal number, or cardinal for short, is what is commonly called the number of elements of a set. In the case of a finite set, its cardinal number, or cardinality is therefore a natural number. For dealing with the cas ...
) of a finite set is equal to the order type. Counting in the everyday sense typically starts from one, so it assigns to each object the size of the initial segment with that object as last element. Note that these numbers are one more than the formal ordinal numbers according to the isomorphic order, because these are equal to the number of earlier objects (which corresponds to counting from zero). Thus for finite , the expression "-th element" of a well-ordered set requires context to know whether this counts from zero or one. In an expression "-th element" where can also be an infinite ordinal, it will typically count from zero. For an infinite set, the order type determines the cardinality, but not conversely: sets of a particular infinite cardinality can have well-orders of many different types (see , below, for an example). For a countably infinite set, the set of possible order types is uncountable.


Examples and counterexamples


Natural numbers

The standard ordering ≤ of the
natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
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: :\begin 0 & 2 & 4 & 6 & 8 & \dots & 1 & 3 & 5 & 7 & 9 & \dots \end This is a well-ordered set of order type . 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 In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
s, the standard ordering ≤ of the
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s is not a well ordering, since, for example, the set of negative integers does not contain a least element. The following binary relation is an example of well ordering of the integers: if and only if one of the following conditions holds: # # is positive, and is negative # and are both positive, and # and are both negative, and This relation can be visualized as follows: :\begin 0 & 1 & 2 & 3 & 4 & \dots & -1 & -2 & -3 & \dots \end is isomorphic to the
ordinal number In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the leas ...
. Another relation for well ordering the integers is the following definition: x \leq_z y if and only if :, x, < , y, \qquad \text \qquad , x, = , y, \text x \leq y. This well order can be visualized as follows: :\begin 0 & -1 & 1 & -2 & 2 & -3 & 3 & -4 & 4 & \dots \end This has the order type .


Reals

The standard ordering ≤ of any real interval is not a well ordering, since, for example, the
open interval In mathematics, a real interval is the set (mathematics), set of all real numbers lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative infinity, indicating the interval extends without ...
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. 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 is a subset of well ordered by . For each in , let be the successor of in ordering on (unless is the last element of ). Let A = \ whose elements are nonempty and disjoint intervals. Each such interval contains at least one rational number, so there is an injective function from to There is an injection from to (except possibly for a last element of , which could be mapped to zero later). And it is well known that there is an injection from to the natural numbers (which could be chosen to avoid hitting zero). Thus there is an injection from to the natural numbers, which means that is countable. On the other hand, a countably infinite subset of the reals may or may not be a well order with the standard . For example, * The natural numbers are a well order under the standard ordering . * The set \ has no least element and is therefore not a well order under standard ordering . Examples of well orders: *The set of numbers \ has order type . *The set of numbers \ has order type . The previous set is the set of limit points 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 \ \cup \ has order type . 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.


Equivalent formulations

If a set is totally ordered, then the following are equivalent to each other: # The set is well ordered. That is, every nonempty subset has a least element. # Transfinite induction works for the entire ordered set. # Every strictly decreasing sequence of elements of the set must terminate after only finitely many steps (assuming the
axiom of dependent choice In mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. T ...
). # Every subordering is isomorphic to an initial segment.


Order topology

Every well-ordered set can be made into a
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
by endowing it with the order topology. With respect to this topology there can be two kinds of elements: * isolated points — these are the minimum and the elements with a predecessor. * limit points — this type does not occur in finite sets, and may or may not occur in an infinite set; the infinite sets without limit point are the sets of order type , for example the natural numbers For subsets we can distinguish: *Subsets with a maximum (that is, subsets that are bounded by themselves); this can be an isolated point or a limit point of the whole set; in the latter case it may or may not be also a limit point of the subset. *Subsets that are unbounded by themselves but bounded in the whole set; they have no maximum, but a supremum outside the subset; if the subset is non-empty this supremum is a limit point of the subset and hence also of the whole set; if the subset is empty this supremum is the minimum of the whole set. *Subsets that are unbounded in the whole set. A subset is cofinal in the whole set if and only if it is unbounded in the whole set or it has a maximum that is also maximum of the whole set. A well-ordered set as topological space is a first-countable space if and only if it has order type less than or equal to ω1 ( omega-one), that is, if and only if the set is
countable In mathematics, a Set (mathematics), set is countable if either it is finite set, finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function fro ...
or has the smallest uncountable order type.


See also

* Tree (set theory), generalization *
Ordinal number In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the leas ...
* Well-founded set * Well partial order * Prewellordering *
Directed set In mathematics, a directed set (or a directed preorder or a filtered set) is a preordered set in which every finite subset has an upper bound. In other words, it is a non-empty preordered set A such that for any a and b in A there exists c in A wit ...


References

* {{Order theory Order theory Ordinal numbers Wellfoundedness