Encyclopedia
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.
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 . Given a class of ordinals, one can identify the a-th member of that class, i.e. one can index 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 . To define things briefly, a well-ordered set is a totally ordered set in which there is no infinite
decreasing sequence . Ordinals may be used to label the elements of any given well-ordered set 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 to 41 , and it is generally identified as the set . 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 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 ?, which is the order type of the natural numbers and which can even be identified with the
set of 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, ?, and after that come ?+1, ?+2, ?+3, and so on. After all of these come ?·2 , ?·2+1, ?·2+2, and so on, then ?·3, and then later on ?·4. Now the set of ordinals which we form in this way must itself have an ordinal associated to it: and that is ?
2. Further on, there will be ?
3, then ?
4, and so on, and ?
?, then ?
?², and much later on e
0 . We can go on in this way indefinitely far .
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 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 . If the states of a computation 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 . 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 . 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 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 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
.
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 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.
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 = , and 2 is equal to and so it is a subset of .
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. . The class of all ordinals is variously called "Ord", "ON", or "8".
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. . 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 is used in showing that these ordinals are well ordered by containment .
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 a to a itself, is true of all ordinals.
That is, if
P is true whenever
P is true for all
ß<
a, then
P is true for
all a. Or, more practically: in order to prove a property
P for all ordinals
a, one can assume that it is already known for all smaller
ß<
a.
Transfinite recursion
Transfinite induction can be used not only to prove things, but also to define them : the formal statement is tedious to write, but the bottom line is, in order to define a function on the ordinals
a, one can assume that it is already defined for all smaller
ß<
a. One proves by transfinite induction that there is one and only one function satisfying the recursion formula up to and including a.
Here is an example of definition by transfinite induction on the ordinals : define a function
F by letting
F be the smallest ordinal not in the set of
F for all
ß<
a. Note how we assume the
F known in the very process of defining
F: this apparent paradox is exactly what definition by transfinite induction permits. Now in fact
F makes sense since there is no
ß<0, so the set of all
F for
ß<0 is empty, so
F must be 0 , and now that we know
F, then
F makes sense , and so on . Well, it turns out that this example is not very interesting since
F=
a for all ordinals
a: 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 ?+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 a, then it is the next ordinal after a, and it is called a
successor ordinal, namely the successor of a, written a+1. In the von Neumann definition of ordinals, the successor of a is since its elements are those of a and a 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 .
When is a sequence of ordinals indexed by a limit ? and the sequence is
increasing, i.e. whenever we define its
limit to be the least upper bound of the set that is, the smallest ordinal greater than any term of the sequence. In this sense, a limit ordinal is the limit of all smaller ordinals .
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 we can find another ordinal larger than it, but still less than ω.
Thus, every ordinal is either zero, or a successor , 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, and
F assuming
F is defined, and then, for limit ordinals d one defines
F as the limit of the
F for all ß
Indexing classes of ordinals
We have mentioned that any well-ordered set is similar to a unique ordinal number , or, in other words, that its elements can be indexed in increasing fashion by the ordinals less than . This applies, in particular, to any set of ordinals: any set of ordinals is naturally indexed by the ordinals less than some . The same holds, with a slight modification, for classes of ordinals : any class of ordinals can be indexed by ordinals . So we can freely speak of the -th element in the class . Formally, the definition is by transfinite induction: the -th element of the class is defined , as the smallest element greater than the -th element for all .
We can apply this, for example, to the class of limit ordinals: the -th ordinal which is either a limit or zero is . Similarly, we can consider ordinals which are additively indecomposable : the -th additively indecomposable ordinal is indexed as . The technique of indexing classes of ordinals is often useful in the context of fixed points: for example, the -th ordinal such that is written . 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 . 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 function is continuous in the sense that, for a limit ordinal, is the limit of all for ; this is also the same as being closed, in the topological sense, for the order topology .
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 . The class of additively indecomposable ordinals, or the class of 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 . 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 ? with the class of ordinals with uncountable cofinality.
Rather than formulating these definitions for classes of ordinals, we can formulate them for sets of ordinals below a given ordinal : A subset of a limit ordinal is said to be unbounded under provided any ordinal less than is less than some ordinal in the set. More generally, we can call a subset of any ordinal cofinal in provided every ordinal less than is less than or equal to some ordinal in the set. The subset is said to be closed under provided it is closed for the order topology in , i.e. a limit of ordinals in the set is either in the set or equal to itself.
Arithmetic of ordinals
There are three usual operations on ordinals: addition, multiplication, and 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 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 a-th infinite initial ordinal is written . Its cardinality is written . For example, the cardinality of ?0 = ? is , which is also the cardinality of ?² or e0 . So we identify ? with , except that the notation is used when writing cardinals, and ? when writing ordinals . Also, is the smallest uncountable ordinal , is the smallest ordinal whose cardinality is greater than , and so on, and is the limit of the for natural numbers n .
Cofinality
The cofinality of an ordinal is the smallest ordinal which is the order type of a cofinal subset of . 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 -indexed strictly increasing sequence with limit . For example, the cofinality of ?² is ?, because the sequence ?·m tends to ?²; but, more generally, any countable limit ordinal has cofinality ?. An uncountable limit ordinal may have either cofinality ? as does 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 .
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 is regular for each a. In this case, the ordinals 0, 1, , , and are regular, whereas 2, 3, , and ??·2 are initial ordinals which are not regular.
The cofinality of any ordinal a is a regular ordinal, i.e. the cofinality of the cofinality of a is the same as the cofinality of a. So the cofinality operation is idempotent.
Some “large” countable ordinals
We have already mentioned the ordinal e0, which is the smallest satisfying the equation , so it is the limit of the sequence 0, 1, , , , etc. Many ordinals can be defined in such a manner as fixed points of certain ordinal functions . 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, , which is the smallest ordinal which cannot in any way be represented by a computable function . Considerably large ordinals can be defined below , however, which measure the “proof-theoretic strength” of certain formal systems . Large ordinals can also be defined above the Church-Kleene ordinal, which are of interest in various parts of logic.
Topology and ordinals
Ordinals as topological spaces
Any ordinal can be made into a topological space by endowing it with the order topology : in the absence of indication to the contrary, it is always that order topology which is meant when an ordinal is thought of as a topological space.
The set of limit points of an ordinal a is precisely the set of limit ordinals less than a. Successor ordinals less than a are isolated points in a. In particular, the finite ordinals and ? are discrete topological spaces, and no ordinal beyond that is discrete. The ordinal a is compact as a topological space if and only if a is a successor ordinal.
The closed sets of a limit ordinal a are just the closed sets in the sense that we have already defined, namely, those which contain a limit ordinal whenever they contain all sufficiently large ordinals below it.
Any ordinal is, of course, an open subset of any further ordinal. We can also define the topology on the ordinals in the following inductive way: 0 is the empty topological space, a+1 is obtained by taking the one-point compactification of a , and for d a limit ordinal, d is equipped with the inductive limit topology.
As topological spaces, all the ordinals are Hausdorff and even normal. They are also totally disconnected , scattered , zero-dimensional . However, they are not extremally disconnected in general .
The topological spaces ?1 and its successor ?1+1 are frequently used as text-book examples of non-countable topological spaces.
For example, in the topological space ?1+1, the element ?1 is in the closure of the subset ?1 even though no sequence of elements in ?1 has the element ?1 as its limit. The space ?1 is first-countable, but not second-countable, and ?1+1 has neither of these two properties, despite being compact. It is also worthy of note that any continuous function from ?1 to R is eventually constant: so the Stone-Cech compactification of ?1 is ?1+1, just as its one-point compactification .
Ordinal-indexed sequences
If a is a limit ordinal and X is a set, an a-indexed sequence of elements of X merely means a function from a to X. If X is a topological space, we say that an a-indexed sequence of elements of X converges to a limit x when it converges as a net, in other words, when given any neighborhood U of x there is an ordinal ßx? is in U for all ?=ß. This coincides with the notion of limit defined above for increasing ordinal-indexed sequences in an ordinal.
Ordinal-indexed sequences are more powerful than ordinary sequences to determine limits in topology: for example, ?1 is a limit point of ?1+1 , and, indeed, it is the limit of the ?1-indexed sequence which maps any ordinal less than ?1 to itself: however, it is not the limit of any ordinary sequence in ?1, since any function from the natural numbers to ?1 is bounded. However, ordinal-indexed sequences are not powerful enough to replace nets in general: for example, on the Tychonoff plank , the corner point is a limit point of the open subset , but it is not the limit of an ordinal-indexed sequence.
See also
- successor ordinal
- limit ordinal
- ordinal arithmetic
- large countable ordinals
- cardinal number
References
- Conway, J. H. and Guy, R. K. "Cantor's Ordinal Numbers." In The Book of Numbers. New York: Springer-Verlag, pp. 266-267 and 274, 1996.
- Sierpinski, W. . Cardinal and Ordinal Numbers . Warszawa: Panstwowe Wydawnictwo Naukowe. Also defines ordinal operations in terms of the Cantor Normal Form.
- Patrick Suppes, Axiomatic Set Theory, D.Van Nostrand Company Inc., 1960, ISBN 0486616304
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