Well-order

In mathematics, a well-order (or well-ordering or well-order relation) on a set Template:Mvar is a total ordering on Template:Mvar with the property that every non-empty subset of Template:Mvar has a least element in this ordering. The set Template:Mvar together with the ordering is then called a well-ordered set (or woset).<ref name="woset">Template:Cite conference</ref> 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 Template:Mvar of a well-ordered set, except a possible greatest element, has a unique successor (next element), namely the least element of the subset of all elements greater than Template:Mvar. There may be elements, besides the least element, that have no predecessor (see Template:Slink below for an example). A well-ordered set Template:Mvar contains for every subset Template:Mvar with an upper bound a least upper bound, namely the least element of the subset of all upper bounds of Template:Mvar in Template:Mvar.

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, 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

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Every well-ordered set is uniquely order isomorphic to a unique ordinal number, 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) of a finite set is equal to the order type.<ref>Template:Cite journal</ref> 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 Template:Mvar, the expression "Template:Mvar-th element" of a well-ordered set requires context to know whether this counts from zero or one. In an expression "Template:Mvar-th element" where Template:Mvar 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 Template:Slink, 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 numbers 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 Template:Math. 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 numbers, the standard ordering ≤ of the integers is not a well ordering, since, for example, the set of negative integers does not contain a least element.

The following binary relation Template:Mvar is an example of well ordering of the integers: Template:Mvar if and only if one of the following conditions holds:

1. Template:Math
2. Template:Mvar is positive, and Template:Mvar is negative
3. Template:Mvar and Template:Mvar are both positive, and Template:Math
4. Template:Mvar and Template:Mvar are both negative, and Template:Math

This relation Template:Mvar can be visualized as follows:

<math>\begin{matrix}

0 & 1 & 2 & 3 & 4 & \dots & -1 & -2 & -3 & \dots \end{matrix}</math> Template:Mvar is isomorphic to the ordinal number Template:Math.

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 Template:Mvar.

Reals

The standard ordering ≤ of any real interval is not a well ordering, since, for example, the open interval Template:Tmath 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>Template:Cite journal</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 Template:Mvar is a subset of Template:Tmath well ordered by Template:Char. For each Template:Mvar in Template:Mvar, let Template:Math be the successor of Template:Mvar in Template:Char ordering on Template:Mvar (unless Template:Mvar is the last element of Template:Mvar). 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 Template:Mvar to Template:Tmath There is an injection from Template:Mvar to Template:Mvar (except possibly for a last element of Template:Mvar, which could be mapped to zero later). And it is well known that there is an injection from Template:Tmath to the natural numbers (which could be chosen to avoid hitting zero). Thus there is an injection from Template:Mvar to the natural numbers, which means that Template:Mvar is countable. On the other hand, a countably infinite subset of the reals may or may not be a well order with the standard Template:Char. For example,

• The natural numbers are a well order under the standard ordering Template: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 Template:Char.

Examples of well orders:

• The set of numbers <math>\{-2^{-n} \,|\, 0 \leq n < \omega\}</math> has order type Template:Math.
• The set of numbers <math>\{-2^{-n} - 2^{-m-n} \,|\, 0 \leq m,n < \omega \}</math> has order type Template:Math. 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 <math>\{-2^{-n} \,|\, 0 \leq n < \omega\} \cup \{1\}</math> has order type Template:Math. 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:

1. The set is well ordered. That is, every nonempty subset has a least element.
2. Transfinite induction works for the entire ordered set.
3. Every strictly decreasing sequence of elements of the set must terminate after only finitely many steps (assuming the axiom of dependent choice).
4. Every subordering is isomorphic to an initial segment.

Order topology

Every well-ordered set can be made into a topological space 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 Template:Mvar, for example the natural numbers Template:Tmath

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 or has the smallest uncountable order type.

See also

• Tree (set theory), generalization
• Ordinal number
• Well-founded set
• Well partial order
• Prewellordering
• Directed set

References

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• Template:Cite book