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).
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 ω2. Further on, there will be ω3, then ω4, and so on, and ωω, then ωω², and much later on ε0 (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 \(\alpha\), or, in other words, that its elements can be indexed in increasing fashion by the ordinals less than \(\alpha\). This applies, in particular, to any set of ordinals: any set of ordinals is naturally indexed by the ordinals less than some \(\alpha\). 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 \(\gamma\)-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 \(\gamma\)-th element of the class is defined (provided it has already been defined for all \(\beta<\gamma\)), as the smallest element greater than the \(\beta\)-th element for all \(\beta<\gamma\).
We can apply this, for example, to the class of limit ordinals: the \(\gamma\)-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 \(\gamma\)-th additively indecomposable ordinal is indexed as \(\omega^\gamma\). The technique of indexing classes of ordinals is often useful in the context of fixed points: for example, the \(\gamma\)-th ordinal \(\alpha\) such that \(\omega^\alpha=\alpha\) 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 \(\delta\) a limit ordinal, \(F(\delta)\) (the \(\delta\)-th ordinal in the class) is the limit of all \(F(\gamma)\) for \(\gamma<\delta\); 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 \(\alpha\): A subset of a limit ordinal \(\alpha\) is said to be unbounded (or cofinal) under \(\alpha\) provided any ordinal less than \(\alpha\) is less than some ordinal in the set. More generally, we can call a subset of any ordinal \(\alpha\) cofinal in \(\alpha\) provided every ordinal less than \(\alpha\) is less than or equal to some ordinal in the set. The subset is said to be closed under \(\alpha\) provided it is closed for the order topology in \(\alpha\), i.e. a limit of ordinals in the set is either in the set or equal to \(\alpha\) itself.
Arithmetic of ordinals
- For more details on this topic, see ordinal arithmetic.
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 \(\omega_\alpha\). Its cardinality is written \(\aleph_\alpha\). For example, the cardinality of ω0 = ω is \(\aleph_0\), which is also the cardinality of ω² or ε0 (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 \(\omega^2>\omega\)). Also, \(\omega_1\) 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 \(\omega_1\) is the order type of that set), \(\omega_2\) is the smallest ordinal whose cardinality is greater than \(\aleph_1\), and so on, and \(\omega_\omega\) is the limit of the \(\omega_n\) for natural numbers n (any limit of cardinals is a cardinal, so this limit is indeed the first cardinal after all the \(\omega_n\)).
Cofinality
The cofinality of an ordinal \(\alpha\) is the smallest ordinal \(\delta\) which is the order type of a cofinal subset of \(\alpha\). 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 \(\delta\)-indexed strictly increasing sequence with limit \(\alpha\). 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 \(\omega_\omega\) 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 \(\omega\).
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 \(\omega_{\alpha+1}\) is regular for each α. In this case, the ordinals 0, 1, \(\omega\), \(\omega_1\), and \(\omega_2\) are regular, whereas 2, 3, \(\omega_\omega\), and ωω·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
- For more details on this topic, see Large countable ordinals.
We have already mentioned (see Cantor normal form) the ordinal ε0, which is the smallest satisfying the equation \(\omega^\alpha = \alpha\), so it is the limit of the sequence 0, 1, \(\omega\), \(\omega^\omega\), \(\omega^{\omega^\omega}\), etc. Many ordinals can be defined in such a manner as fixed points of certain ordinal functions (the \(\iota\)-th ordinal such that \(\omega^\alpha = \alpha\) is called \(\varepsilon_\iota\), then we could go on trying to find the \(\iota\)-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 \(\omega_1\) 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
- For more details on this topic, see Order topology.
Any ordinal can be made into a topological space in a natural way by endowing it with the order topology. See the Topology and ordinals section of the "Order topology" article.
See also
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.
- Sierpiński, W. (1965). Cardinal and Ordinal Numbers (2nd ed.). Warszawa: Państwowe 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 0-486-61630-4