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Ordinal number

2007 Schools Wikipedia Selection. Related subjects: Mathematics

   Commonly, ordinal numbers, or ordinals for short, are numbers used to
   denote the position in an ordered sequence: first, second, third,
   fourth, etc., whereas a cardinal number says "how many there are": one,
   two, three, four, etc. (See How to name numbers.)

   Here, we describe the mathematical meaning of transfinite ordinal
   numbers. They were introduced by Georg Cantor in 1897, to accommodate
   infinite sequences and to classify sets with certain kinds of order
   structures on them. Ordinals are an extension of the natural numbers
   different from integers and from cardinals.

   Well-ordering is total ordering with transfinite induction, where
   transfinite induction extends mathematical induction beyond the finite.
   Ordinals represent equivalence classes of well orderings with
   order-isomorphism being the equivalence relationship. Each ordinal is
   taken to be the set of smaller ordinals. Ordinals may be categorized
   as: zero, successor ordinals, and limit ordinals (of various
   cofinalities). Given a class of ordinals, one can identify the α-th
   member of that class, i.e. one can index (count) them. A class is
   closed and unbounded if its indexing function is continuous and never
   stops. One can define addition, multiplication, and exponentiation on
   ordinals, but not subtraction or division. The Cantor normal form is a
   standardized way of writing down ordinals. There is a many to one
   association from ordinals to cardinals. Larger and larger ordinals can
   be defined, but they become more and more difficult to describe.
   Ordinals have a natural topology.

Ordinals extend the natural numbers

   A natural number can be used for two purposes: to describe the size of
   a set, or to describe the position of an element in a sequence. While
   in the finite world these two concepts coincide, when dealing with
   infinite sets one has to distinguish between the two. The notion of
   size leads to cardinal numbers, which were also discovered by Cantor,
   while the position is generalized by the ordinal numbers described
   here.

   Whereas the notion of cardinal number is associated to a set with no
   particular structure on it, the ordinals are intimately linked with the
   special kind of sets which are called well-ordered (so intimately
   linked, in fact, that some mathematicians make no distinction between
   the two concepts). To define things briefly, a well-ordered set is a
   totally ordered set (given any two elements one defines a smaller and a
   larger one in a coherent way) in which there is no infinite decreasing
   sequence (however, there may be infinite increasing sequences).
   Ordinals may be used to label the elements of any given well-ordered
   set (the smallest element being labeled 0, the one after that 1, the
   next one 2, "and so on") and to measure the "length" of the whole set
   by the least ordinal which is not a label for an element of the set.
   This "length" is called the order type of the set.

   Any ordinal is defined by the set of ordinals that precede it: in fact,
   the most common definition of ordinals identifies each ordinal as the
   set of ordinals that precede it. For example, the ordinal 42 is the
   order type of the ordinals less than it, i.e., the ordinals from 0 (the
   smallest of all ordinals) to 41 (the immediate predecessor of 42), and
   it is generally identified as the set {0,1,2,…,41}. Conversely, any set
   of ordinals which is downward-closed—meaning that any ordinal less than
   an ordinal in the set is also in the set—is (or can be identified with)
   an ordinal.

   So far we have mentioned only finite ordinals, which are the natural
   numbers. But there are infinite ones as well: the smallest infinite
   ordinal is ω, which is the order type of the natural numbers (finite
   ordinals) and which can even be identified with the set of natural
   numbers (indeed, the set of natural numbers is well-ordered—as is any
   set of ordinals—and since it is downward closed it can be identified
   with the ordinal associated to it, which is exactly how we define ω).
   A graphical “matchstick” representation of the ordinal ω². Each stick
   correspond to an ordinal of the form ω·m+n where m and n are natural
   numbers.
   Enlarge
   A graphical “matchstick” representation of the ordinal ω². Each stick
   correspond to an ordinal of the form ω·m+n where m and n are natural
   numbers.

   Perhaps a clearer intuition of ordinals can be formed by examining a
   first few of them: as mentioned above, they start with the natural
   numbers, 0, 1, 2, 3, 4, 5, … After all natural numbers comes the first
   infinite ordinal, ω, and after that come ω+1, ω+2, ω+3, and so on.
   (Exactly what addition means will be defined later on: just consider
   them as names.) After all of these come ω·2 (which is ω+ω), ω·2+1,
   ω·2+2, and so on, then ω·3, and then later on ω·4. Now the set of
   ordinals which we form in this way (the ω·m+n, where m and n are
   natural numbers) must itself have an ordinal associated to it: and that
   is ω^2. Further on, there will be ω^3, then ω^4, and so on, and ω^ω,
   then ω^ω², and much later on ε[0] (just to give a few examples of the
   very smallest—countable—ordinals). We can go on in this way
   indefinitely far ("indefinitely far" is exactly what ordinals are good
   at: basically every time one says "and so on" when enumerating
   ordinals, it defines a larger ordinal).

Definitions

Define well-ordered set

   A well-ordered set is an ordered set in which every non-empty subset
   has a least element: this is equivalent (at least in the presence of
   the axiom of dependent choices) to just saying that the set is totally
   ordered and there is no infinite decreasing sequence, something which
   is perhaps easier to visualize. In practice, the importance of
   well-ordering is justified by the possibility of applying transfinite
   induction, which says, essentially, that any property that passes on
   from the predecessors of an element to that element itself must be true
   of all elements (of the given well-ordered set). If the states of a
   computation (computer program or game) can be well-ordered in such a
   way that each step is followed by a "lower" step, then you can be sure
   that the computation will terminate.

   Now we don't want to distinguish between two well-ordered sets if they
   only differ in the "labeling of their elements", or more formally: if
   we can pair off the elements of the first set with the elements of the
   second set such that if one element is smaller than another in the
   first set, then the partner of the first element is smaller than the
   partner of the second element in the second set, and vice versa. Such a
   one-to-one correspondence is called an order isomorphism and the two
   well-ordered sets are said to be order-isomorphic, or similar
   (obviously this is an equivalence relation). Provided there exists an
   order isomorphism between two well-ordered sets, the order isomorphism
   is unique: this makes it quite justifiable to consider the sets as
   essentially identical, and to seek a "canonical" representative of the
   isomorphism type (class). This is exactly what the ordinals provide,
   and it also provides a canonical labeling of the elements of any
   well-ordered set.

   So we essentially wish to define an ordinal as an isomorphism class of
   well-ordered sets: that is, as an equivalence class for the equivalence
   relation of "being order-isomorphic". There is a technical difficulty
   involved, however, in the fact that the equivalence class is too large
   to be a set in the usual Zermelo–Fraenkel (ZF) formalization of set
   theory. But this is not a serious difficulty. We will say that the
   ordinal is the order type of any set in the class.

Definition of an ordinal as an equivalence class

   The original definition of ordinal number, found for example in
   Principia Mathematica, defines the order type of a well-ordering as the
   set of all well-orderings similar (order-isomorphic) to that
   well-ordering: in other words, an ordinal number is genuinely an
   equivalence class of well-ordered sets. This definition must be
   abandoned in ZF and related systems of axiomatic set theory because
   these equivalence classes are too large to form a set. However, this
   definition still can be used in type theory and in Quine's set theory
   New Foundations and related systems (where it affords a rather
   surprising alternative solution to the Burali-Forti paradox of the
   largest ordinal).

Von Neumann definition of ordinals

   Rather than defining an ordinal as an equivalence class of well-ordered
   sets, we can try to define it as some particular well-ordered set which
   (canonically) represents the class. Thus, we want to construct ordinal
   numbers as special well-ordered sets in such a way that every
   well-ordered set is order-isomorphic to one and only one ordinal
   number.

   The ingenious definition suggested by John von Neumann, and which is
   now taken as standard, is this: define each ordinal as a special
   well-ordered set, namely that of all ordinals before it. Formally:

          A set S is an ordinal if and only if S is totally ordered with
          respect to set containment and every element of S is also a
          subset of S.

   (Here, "set containment" is another name for the subset relationship.)
   Such a set S is automatically well-ordered with respect to set
   containment. This relies on the axiom of well foundation: every
   nonempty set B has an element b which is disjoint from B.

   Note that the natural numbers are ordinals by this definition. For
   instance, 2 is an element of 4 = {0, 1, 2, 3}, and 2 is equal to {0, 1}
   and so it is a subset of {0, 1, 2, 3}.

   It can be shown by transfinite induction that every well-ordered set is
   order-isomorphic to exactly one of these ordinals.

   Furthermore, the elements of every ordinal are ordinals themselves.
   Whenever you have two ordinals S and T, S is an element of T if and
   only if S is a proper subset of T, and moreover, either S is an element
   of T, or T is an element of S, or they are equal. So every set of
   ordinals is totally ordered. And in fact, much more is true: Every set
   of ordinals is well-ordered. This important result generalizes the fact
   that every set of natural numbers is well-ordered and it allows us to
   use transfinite induction liberally with ordinals.

   Another consequence is that every ordinal S is a set having as elements
   precisely the ordinals smaller than S. This statement completely
   determines the set-theoretic structure of every ordinal in terms of
   other ordinals. It is used to prove many other useful results about
   ordinals. One example of these is an important characterization of the
   order relation between ordinals: every set of ordinals has a supremum,
   the ordinal obtained by taking the union of all the ordinals in the
   set. Another example is the fact that the collection of all ordinals is
   not a set. Indeed, since every ordinal contains only other ordinals, it
   follows that every member of the collection of all ordinals is also its
   subset. Thus, if that collection were a set, it would have to be an
   ordinal itself by definition; then it would be its own member, which
   contradicts the axiom of regularity. (See also the Burali-Forti
   paradox). The class of all ordinals is variously called "Ord", "ON", or
   "∞".

   An ordinal is finite if and only if the opposite order is also
   well-ordered, which is the case if and only if each of its subsets has
   a greatest element.

Other definitions

   There are other modern formulations of the definition of ordinal. Each
   of these is essentially equivalent to the definition given above. One
   of these definitions is the following. A class S is called transitive
   if each element x of S is a subset of S, i.e. y \in x \in S
   \Longrightarrow y \in S . An ordinal is then defined to be a transitive
   set whose members are also transitive. It follows from this that the
   members are themselves ordinals. Note that the axiom of regularity
   (foundation) is used in showing that these ordinals are well ordered by
   containment (subset).

Transfinite induction

What is transfinite induction?

   Transfinite induction holds in any well-ordered set, but it is so
   important in relation to ordinals that it is worth restating here.

          Any property which passes from the set of ordinals smaller than
          a given ordinal α to α itself, is true of all ordinals.

   That is, if P(α) is true whenever P(β) is true for all β<α, then P(α)
   is true for all α. Or, more practically: in order to prove a property P
   for all ordinals α, one can assume that it is already known for all
   smaller β<α.

Transfinite recursion

   Transfinite induction can be used not only to prove things, but also to
   define them (such a definition is normally said to follow by
   transfinite recursion - we use transfinite induction to prove that the
   result is well-defined): the formal statement is tedious to write, but
   the bottom line is, in order to define a (class) function on the
   ordinals α, one can assume that it is already defined for all smaller
   β<α. One proves by transfinite induction that there is one and only one
   function satisfying the recursion formula up to and including α.

   Here is an example of definition by transfinite induction on the
   ordinals (more will be given later): define a function F by letting
   F(α) be the smallest ordinal not in the set of F(β) for all β<α. Note
   how we assume the F(β) known in the very process of defining F: this
   apparent paradox is exactly what definition by transfinite induction
   permits. Now in fact F(0) makes sense since there is no β<0, so the set
   of all F(β) for β<0 is empty, so F(0) must be 0 (the smallest ordinal
   of all), and now that we know F(0), then F(1) makes sense (and it is
   the smallest ordinal not equal to F(0)=0), and so on (the and so on is
   exactly transfinite induction). Well, it turns out that this example is
   not very interesting since F(α)=α for all ordinals α: but this can be
   shown, precisely, by transfinite induction.

Successor and limit ordinals

   Any nonzero ordinal has a smallest element, to wit zero. It may or may
   not have a largest element. For example, 42 has a largest element 41
   and ω+6 has a largest element ω+5. On the other hand, ω does not have a
   largest element since there is no largest natural number. If an ordinal
   has a largest element α, then it is the next ordinal after α, and it is
   called a successor ordinal, namely the successor of α, written α+1. In
   the von Neumann definition of ordinals, the successor of α is
   \alpha\cup\{\alpha\} since its elements are those of α and α itself.

   A nonzero ordinal which is not a successor is called a limit ordinal.
   One justification for this term is that a limit ordinal is indeed the
   limit in a topological sense of all smaller ordinals (for the order
   topology).

   When \langle \alpha_{\iota} | \iota < \gamma \rangle is a sequence of
   ordinals indexed by a limit γ and the sequence is increasing, i.e.
   \alpha_{\iota} < \alpha_{\rho}\! whenever \iota < \rho,\! we define its
   limit to be the least upper bound of the set \{ \alpha_{\iota} | \iota
   < \gamma \},\! that is, the smallest ordinal (it always exists) greater
   than any term of the sequence. In this sense, a limit ordinal is the
   limit of all smaller ordinals (indexed by itself).

   Another way of defining a limit ordinal is to say that α is a limit
   ordinal if and only if:

   There is an ordinal less than α and whenever ζ is an ordinal less than
   α, then there exists an ordinal ξ such that ζ<ξ<α.

   So in the following sequence

   0, 1, 2, ... , ω, ω+1

   ω is a limit ordinal because for any ordinal (in this example, a
   natural number) we can find another ordinal (natural number) larger
   than it, but still less than ω.

   Thus, every ordinal is either zero, or a successor (of a well-defined
   predecessor), or a limit. This distinction is important, because many
   definitions by transfinite induction rely upon it. Very often, when
   defining a function F by transfinite induction on all ordinals, one
   defines F(0), and F(α+1) assuming F(α) is defined, and then, for limit
   ordinals δ one defines F(δ) as the limit of the F(β) for all β<δ
   (either in the sense of ordinal limits, as we have just explained, or
   for some other notion of limit if F does not take ordinal values).
   Thus, the interesting step in the definition is the successor step, not
   the limit ordinals. Such functions (especially for F nondecreasing and
   taking ordinal values) are called continuous. We will see that ordinal
   addition, multiplication and exponentiation are continuous as functions
   of their second argument.

Indexing classes of ordinals

   We have mentioned that any well-ordered set is similar
   (order-isomorphic) to a unique ordinal number α, or, in other words,
   that its elements can be indexed in increasing fashion by the ordinals
   less than α. This applies, in particular, to any set of ordinals: any
   set of ordinals is naturally indexed by the ordinals less than some α.
   The same holds, with a slight modification, for classes of ordinals (a
   collection of ordinals, possibly too large to form a set, defined by
   some property): any class of ordinals can be indexed by ordinals (and,
   when the class is unbounded, this puts it in class-bijection with the
   class of all ordinals). So we can freely speak of the γ-th element in
   the class (with the convention that the “0-th” is the smallest, the
   “1-th” is the next smallest, and so on). Formally, the definition is by
   transfinite induction: the γ-th element of the class is defined
   (provided it has already been defined for all β < γ), as the smallest
   element greater than the β-th element for all β < γ.

   We can apply this, for example, to the class of limit ordinals: the
   γ-th ordinal which is either a limit or zero is \omega\cdot\gamma (see
   ordinal arithmetic for the definition of multiplication of ordinals).
   Similarly, we can consider ordinals which are additively indecomposable
   (meaning that it is a nonzero ordinal which is not the sum of two
   strictly smaller ordinals): the γ-th additively indecomposable ordinal
   is indexed as ω^γ. The technique of indexing classes of ordinals is
   often useful in the context of fixed points: for example, the γ-th
   ordinal α such that ω^α = α is written \varepsilon_\gamma . These are
   called the "epsilon numbers".

Closed unbounded sets and classes

   A class of ordinals is said to be unbounded, or cofinal, when given any
   ordinal, there is always some element of the class greater than it
   (then the class must be a proper class, i.e., it cannot be a set). It
   is said to be closed when the limit of a sequence of ordinals in the
   class is again in the class: or, equivalently, when the indexing
   (class-)function F is continuous in the sense that, for δ a limit
   ordinal, F(δ) (the δ-th ordinal in the class) is the limit of all F(γ)
   for γ < δ; this is also the same as being closed, in the topological
   sense, for the order topology (to avoid talking of topology on proper
   classes, one can demand that the intersection of the class with any
   given ordinal is closed for the order topology on that ordinal, this is
   again equivalent).

   Of particular importance are those classes of ordinals which are closed
   and unbounded, sometimes called clubs. For example, the class of all
   limit ordinals is closed and unbounded: this translates the fact that
   there is always a limit ordinal greater than a given ordinal, and that
   a limit of limit ordinals is a limit ordinal (a fortunate fact if the
   terminology is to make any sense at all!). The class of additively
   indecomposable ordinals, or the class of \varepsilon_\cdot ordinals, or
   the class of cardinals, are all closed unbounded; the set of regular
   cardinals, however, is unbounded but not closed, and any finite set of
   ordinals is closed but not unbounded.

   A class is stationary if it has a nonempty intersection with every
   closed unbounded class. All superclasses of closed unbounded classes
   are stationary and stationary classes are unbounded, but there are
   stationary classes which are not closed and there are stationary
   classes which have no closed unbounded subclass (such as the class of
   all limit ordinals with countable cofinality). Since the intersection
   of two closed unbounded classes is closed and unbounded, the
   intersection of a stationary class and a closed unbounded class is
   stationary. But the intersection of two stationary classes may be
   empty, e.g. the class of ordinals with cofinality ω with the class of
   ordinals with uncountable cofinality.

   Rather than formulating these definitions for (proper) classes of
   ordinals, we can formulate them for sets of ordinals below a given
   ordinal α: A subset of a limit ordinal α is said to be unbounded (or
   cofinal) under α provided any ordinal less than α is less than some
   ordinal in the set. More generally, we can call a subset of any ordinal
   α cofinal in α provided every ordinal less than α is less than or equal
   to some ordinal in the set. The subset is said to be closed under α
   provided it is closed for the order topology in α, i.e. a limit of
   ordinals in the set is either in the set or equal to α itself.

Arithmetic of ordinals

   There are three usual operations on ordinals: addition, multiplication,
   and (ordinal) exponentiation. Each can be defined in essentially two
   different ways: either by constructing an explicit well-ordered set
   which represents the operation or by using transfinite recursion. The
   Cantor normal form provides a standardized way of writing ordinals. The
   so-called "natural" arithmetical operations retain commutativity at the
   expense of continuity.

Ordinals and cardinals

Initial ordinal of a cardinal

   Each ordinal has an associated cardinal, its cardinality, obtained by
   simply forgetting the order. Any well-ordered set having that ordinal
   as its order-type has the same cardinality. The smallest ordinal having
   a given cardinal as its cardinality is called the initial ordinal of
   that cardinal. Every finite ordinal (natural number) is initial, but
   most infinite ordinals are not initial. The axiom of choice is
   equivalent to the statement that every set can be well-ordered, i.e.
   that every cardinal has an initial ordinal. In this case, it is
   traditional to identify the cardinal number with its initial ordinal,
   and we say that the initial ordinal is a cardinal.

   The α-th infinite initial ordinal is written ω[α]. Its cardinality is
   written \aleph_\alpha . For example, the cardinality of ω[0] = ω is
   \aleph_0 , which is also the cardinality of ω² or ε[0] (all are
   countable ordinals). So (assuming the axiom of choice) we identify ω
   with \aleph_0 , except that the notation \aleph_0 is used when writing
   cardinals, and ω when writing ordinals (this is important since
   \aleph_0^2=\aleph_0 whereas ω^2 > ω). Also, ω[1] is the smallest
   uncountable ordinal (to see that it exists, consider the set of
   equivalence classes of well-orderings of the natural numbers: each such
   well-ordering defines a countable ordinal, and ω[1] is the order type
   of that set), ω[2] is the smallest ordinal whose cardinality is greater
   than \aleph_1 , and so on, and ω[ω] is the limit of the ω[n] for
   natural numbers n (any limit of cardinals is a cardinal, so this limit
   is indeed the first cardinal after all the ω[n]).

Cofinality

   The cofinality of an ordinal α is the smallest ordinal δ which is the
   order type of a cofinal subset of α. Notice that a number of authors
   define confinality or use it only for limit ordinals. The cofinality of
   a set of ordinals or any other well ordered set is the cofinality of
   the order type of that set.

   Thus for a limit ordinal, there exists a δ-indexed strictly increasing
   sequence with limit α. For example, the cofinality of ω² is ω, because
   the sequence ω·m (where m ranges over the natural numbers) tends to ω²;
   but, more generally, any countable limit ordinal has cofinality ω. An
   uncountable limit ordinal may have either cofinality ω as does ω[ω] or
   an uncountable cofinality.

   The cofinality of 0 is 0. And the cofinality of any successor ordinal
   is 1. The cofinality of any limit ordinal is at least ω.

   An ordinal which is equal to its cofinality is called regular and it is
   always an initial ordinal. Any limit of regular ordinals is a limit of
   initial ordinals and thus is also initial even if it is not regular
   which it usually is not. If the Axiom of Choice, then ω[α + 1] is
   regular for each α. In this case, the ordinals 0, 1, ω, ω[1], and ω[2]
   are regular, whereas 2, 3, ω[ω], and ω[ω·2] are initial ordinals which
   are not regular.

   The cofinality of any ordinal α is a regular ordinal, i.e. the
   cofinality of the cofinality of α is the same as the cofinality of α.
   So the cofinality operation is idempotent.

Some “large” countable ordinals

   We have already mentioned (see Cantor normal form) the ordinal ε[0],
   which is the smallest satisfying the equation ω^α = α, so it is the
   limit of the sequence 0, 1, ω, ω^ω, \omega^{\omega^\omega} , etc. Many
   ordinals can be defined in such a manner as fixed points of certain
   ordinal functions (the ι-th ordinal such that ω^α = α is called
   \varepsilon_\iota , then we could go on trying to find the ι-th ordinal
   such that \varepsilon_\alpha = \alpha , “and so on”, but all the
   subtlety lies in the “and so on”). We can try to do this
   systematically, but no matter what system is used to define and
   construct ordinals, there is always an ordinal that lies just above all
   the ordinals constructed by the system. Perhaps the most important
   ordinal which limits in this manner a system of construction is the
   Church- Kleene ordinal, \omega_1^{\mathrm{CK}} (despite the ω[1] in the
   name, this ordinal is countable), which is the smallest ordinal which
   cannot in any way be represented by a computable function (this can be
   made rigorous, of course). Considerably large ordinals can be defined
   below \omega_1^{\mathrm{CK}} , however, which measure the
   “proof-theoretic strength” of certain formal systems (for example,
   \varepsilon_0 measures the strength of Peano arithmetic). Large
   ordinals can also be defined above the Church-Kleene ordinal, which are
   of interest in various parts of logic.

Topology and ordinals

   Any ordinal can be made into a topological space in a natural way by
   endowing it with the order topology. See Order topology#Topology and
   ordinals.

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