well-orderable set

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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a well-order (or well-ordering or well-order relation) on a set ''S'' is a
total order In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive) ...
on ''S'' with the property that every non-empty
subset In mathematics, set ''A'' is a subset of a set ''B'' if all 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 are unequal, then ''A'' is a proper subset o ...
of ''S'' has a
least 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 dually, that is, it is an ele ...
in this ordering. The set ''S'' together with the well-order relation is then called a well-ordered set. 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 ''s'' 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 dually, that is, it is an el ...
, has a unique successor (next element), namely the least element of the subset of all elements greater than ''s''. There may be elements besides the least element which have no predecessor (see below for an example). A well-ordered set ''S'' contains for every subset ''T'' with an upper bound a
least upper bound In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest ...
, namely the least element of the subset of all upper bounds of ''T'' in ''S''. 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 least ...
, called the order type of the well-ordered set. The well-ordering theorem, which is equivalent to the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
, states that every set can be well ordered. If a set is well ordered (or even if it merely admits a
well-founded relation In mathematics, a binary relation ''R'' is called well-founded (or wellfounded) on a class ''X'' if every non-empty subset ''S'' ⊆ ''X'' has a minimal element with respect to ''R'', that is, an element ''m'' not related by ''s  ...
), the proof technique of
transfinite induction Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers. Its correctness is a theorem of ZFC. Induction by cases Let P(\alpha) be a property defined for ...
can be used to prove that a given statement is true for all elements of the set. The observation that the
natural numbers In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal n ...
are well ordered by the usual less-than relation is commonly called the
well-ordering principle In mathematics, the well-ordering principle states that every non-empty set of positive integers contains a least element. In other words, the set of positive integers is well-ordered by its "natural" or "magnitude" order in which x precedes y ...
(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 least ...
, 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 Counting is the process of determining the number of elements of a finite set of objects, i.e., determining the size of a set. The traditional way of counting consists of continually increasing a (mental or spoken) counter by a unit for every ele ...
, 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, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. Th ...
) 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 ''n'', the expression "''n''-th element" of a well-ordered set requires context to know whether this counts from zero or one. In a notation "β-th element" where β can also be an infinite ordinal, it will typically count from zero. For an infinite set the order type determines the
cardinality In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
, but not conversely: well-ordered sets of a particular cardinality can have many different order types (see , below, for an example). For a
countably infinite In mathematics, a set is countable if either it is 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 from it into the natural number ...
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 those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
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: :0 2 4 6 8 ... 1 3 5 7 9 ... This is a well-ordered set of order type ω + ω. 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 those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
s, 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 relation ''R'' is an example of well ordering of the integers: '' x R y''
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...
one of the following conditions holds: # ''x'' = 0 # ''x'' is positive, and ''y'' is negative # ''x'' and ''y'' are both positive, and ''x'' ≤ ''y'' # ''x'' and ''y'' are both negative, and , ''x'', ≤ , ''y'', This relation ''R'' can be visualized as follows: :0 1 2 3 4 ... −1 −2 −3 ... ''R'' 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 least ...
ω + ω. Another relation for well ordering the integers is the following definition: ''x'' ≤z ''y''
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...
(, ''x'',  < , ''y'', or (, ''x'',  = , ''y'', and ''x'' ≤ ''y'')). This well order can be visualized as follows: : 0 −1 1 −2 2 −3 3 −4 4 ... This has the order type ω.

## 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 a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Othe ...
(0, 1) ⊆ ,1does not contain a least element. From the ZFC axioms of set theory (including the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
) 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 ''X'' is a subset of R well ordered by ≤. For each ''x'' in ''X'', let ''s''(''x'') be the successor of ''x'' in ≤ ordering on ''X'' (unless ''x'' is the last element of ''X''). Let ''A'' = whose elements are nonempty and disjoint intervals. Each such interval contains at least one rational number, so there is an
injective function In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contraposit ...
from ''A'' to Q. There is an injection from ''X'' to ''A'' (except possibly for a last element of ''X'' which could be mapped to zero later). And it is well known that there is an injection from ''Q'' to the natural numbers (which could be chosen to avoid hitting zero). Thus there is an injection from ''X'' to the natural numbers which means that ''X'' 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 ω. *The set of numbers has order type ω2. 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 ∪ has order type ω + 1. With the
order topology In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets. If ''X'' is a totally ordered set, th ...
of this set, 1 is a limit point of the set. With the ordinary topology (or equivalently, the order topology) of the real numbers it is not.

# Equivalent formulations

If a set is
totally ordered In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexiv ...
, then the following are equivalent to each other: # The set is well ordered. That is, every nonempty subset has a least element. #
Transfinite induction Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers. Its correctness is a theorem of ZFC. Induction by cases Let P(\alpha) be a property defined for ...
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, the axiom of dependent choice, denoted by \mathsf , is a weak form of the axiom of choice ( \mathsf ) that is still sufficient to develop most of real analysis. It was introduced by Paul Bernays in a 1942 article that explores wh ...
). # 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 In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets. If ''X'' is a totally ordered set, th ...
. 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 ω, for example N. For subsets we can distinguish: *Subsets with a maximum (that is, subsets which 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 which 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 which 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 which 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 is countable if either it is 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 from it into the natural numbe ...
or has the smallest
uncountable In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal nu ...
order type.