Ordinal number

For other senses of this term, see ordinal (disambiguation).

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.)

In mathematics, ordinal numbers are an extension of the natural numbers, introduced by Georg Cantor in 1897, to accommodate infinite sequences and to classify sets with certain kinds of order structures on them. It is this generalization which will be explained below.

Introduction

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.

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 ω^ω, then ω^ω², 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).

Ordinals and well-ordered sets

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 one 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 (or a strictly increasing function) 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 provides, 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 (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 S has an element a which is disjoint from S.

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's 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 transitive if, whenever x is an element of y and y is an element of S, then x is an element of S. An ordinal is then defined to be a class S which is transitive, and such that every member of S is also transitive. Note that this definition will only work in the presence of the axiom of regularity (foundation): a set which is its own sole element will satisfy this condition and is not an ordinal!

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 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.

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 (which is zero). It may or may not have a largest element, however: 42 or ω+6 have a largest element, whereas ω does not (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).

Quite generally, when (α_ι<γ) is a sequence of ordinals (a family indexed by a limit γ), and if we assume that (α_ι) is increasing (α_ι<α_ι′ whenever ι<ι′), or at any rate non-decreasing, we define its limit to be the least upper bound of the set {α_ι}, 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).

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, on 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 $\omega \cdot \gamma$ (so far we have not defined multiplication but we can take this notation as a temporary definition, which will agree with the general notion to be defined later). 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 such that $\omega ^{\alpha }=\alpha$ is written $\varepsilon _{\gamma }$.

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

There are three usual operations on ordinals: addition, multiplication, and (ordinal) exponentiation. Each can be defined in essentially two different ways: either by using transfinite recursion, or by constructing an explicit well-ordered set which represents the operation.

Addition

The union of two disjoint well-ordered sets S and T can be well-ordered. The order-type of that union is the ordinal which results from adding the order-types of S and T. If two well-ordered sets are not already disjoint, then they can be replaced by order-isomorphic disjoint sets, i.e. replace S by S×{0} and T by T×{1}. Thus the well-ordered set S is written "to the left" of the well-ordered set T, meaning one defines an order on S$\cup$T in which every element of S is smaller than every element of T. The sets S and T themselves keep the ordering they already have. This addition is associative and generalizes the addition of natural numbers.

The first transfinite ordinal is ω, the set of all natural numbers. Let's try to visualize the ordinal ω+ω: two copies of the natural numbers ordered in the normal fashion and the second copy completely to the right of the first. If we write the second copy as {0'<1'<2',...} then ω+ω looks like

0 < 1 < 2 < 3 < ... < 0' < 1' < 2' < ...

This is different from ω because in ω only 0 does not have a direct predecessor while in ω+ω the two elements 0 and 0' don't have direct predecessors. Here are 3 + ω and ω + 3:

0 < 1 < 2 < 0' < 1' < 2' < ...

0 < 1 < 2 < ... < 0' < 1' < 2'

After relabeling, the former just looks like ω itself while the latter doesn't: we have 3 + ω = ω. But ω + 3 is not equal to ω since the former has a largest element and the latter doesn't. So our addition is not commutative.

One can see for example that (ω + 4) + ω = ω + (4 + ω) = ω + ω.

The definition of addition can also be given inductively (the following induction is on β):

• α + 0 = α,
• α + (β+1) = (α+β)+1 (here, "+1" denotes the successor of an ordinal),
• and if δ is limit then α+δ is the limit of the α+β for all β<δ.

Using this definition, we also see that ω+3 is a successor ordinal (it is the successor of ω+2) whereas 3+ω is the limit of 3+0=3, 3+1=4, 3+2=5, etc., which is just ω.

Zero is an additive identity α + 0 = 0 + α = 0.

Addition is associative (α + β) + γ = α + (β + γ).

Addition is strictly increasing and continuous in the right argument:

$\alpha <\beta \Longrightarrow \gamma +\alpha <\gamma +\beta$

but the analogous relation does not hold for the left argument; instead we have:

$\alpha \leq \beta \Longrightarrow \alpha +\gamma \leq \beta +\gamma$

Unrestricted subtraction cannot be defined for ordinals. However, there is a cancellation law from the left: If α + β = α + γ, then β = γ. On the other hand, cancellation from the right does not work:

$3+\omega =0+\omega =\omega$ but $3\neq 0$

Also if β ≤ α, then there is a unique γ such that α = β + γ.

Multiplication

The Cartesian Product, S×T, of two well-ordered sets S and T can be well-ordered (by a variant of lexicographical order which puts the least significant position first). Effectively, each element of T is replaced by a disjoint copy of S. The order-type of the Cartesian Product is the ordinal which results from multiplying the order-types of S and T. Again, this operation is associative and generalizes the multiplication of natural numbers.

Here is ω·2:

0₀ < 1₀ < 2₀ < 3₀ < ... < 0₁ < 1₁ < 2₁ < 3₁ < ...

and we see: ω·2 = ω + ω. But 2·ω looks like this:

0₀ < 1₀ < 0₁ < 1₁ < 0₂ < 1₂ < 0₃ < 1₃ < ...

and after relabeling, this looks just like ω and so we get 2·ω = ω. Multiplication of ordinals is not commutative.

Distributivity partially holds for ordinal arithmetic: R(S+T) = RS + RT. However, the other distributive law (T+U)R = TR + UR is not generally true: (1 + 1)·ω = 2·ω = ω while 1·ω + 1·ω = ω+ω which is different. Therefore, the ordinal numbers do not form a ring.

The definition of multiplication can also be given inductively (the following induction is on β):

• α · 0 = 0,
• α · (β+1) = (α·β)+α,
• and if δ is limit then α·δ is the limit of the α·β for all β<δ.

The main properties of the product are:

α·0 = 0·α = 0.

One is a multiplicative identity α·1 = 1·α = α.

Multiplication is associative (α·β)·γ = α·(β·γ).

Multiplication is strictly increasing and continuous in the right argument:

(α < β and γ > 0) $\Longrightarrow$ γ·α < γ·β,

but if one reverses the arguments the inequality does not work:

for example, 1 < 2 but 1·ω = 2·ω = ω.

Instead one gets α ≤ β $\Longrightarrow$ α·γ ≤ β·γ.

There is a cancellation law: If α > 0 and α · β = α · γ, then β = γ.

The example above shows that cancellation from the right does not work.

α·β = 0 $\Longrightarrow$ α = 0 or β = 0.

α·(β + γ) = α·β + α·γ (distributive law on the left). On the other hand, there is no distributive law on the right.

e.g. (ω + 1)·2 = ω + 1 + ω + 1 = ω + ω + 1 = ω·2 + 1 which is not ω·2 + 2.

Also for all α and β, if β > 0, then there are unique γ and δ such that α = β · γ + δ and δ < β (a kind of euclidian division; this doesn't mean the ordinals are a euclidian domain, however, since they aren't even a ring).

Exponentiation

We can define exponentiation of well ordered sets as follows. If the exponent is a finite set, this should be obvious, for instance ω² = ω·ω using the operation of ordinal multiplication. But instead this can be visualized as the set of functions from 2 = {0,1} to ω = the natural numbers, ordered by a variant of lexicographical order which puts the least significant position first:

(0,0) < (1,0) < (2,0) < (3,0) < ... < (0,1) < (1,1) < (2,1) < (3,1) < ... < (0,2) < (1,2) < (2,2) < ...

Where for brevity, we have replaced the function {(0,i),(1,j)} by the ordered pair (i,j).

And similarly for any finite exponent n, $\omega ^{n}$ can be visualized as the set of functions from n (the domain) to the natural numbers (the range), those functions can be abbreviated as n-tuples of natural numbers.

Then for $\omega ^{\omega }$, we might try to visualize the set of infinite sequences of natural numbers. However, if we try to use any variant of lexicographical order on this set, we find it is not well-ordered. We must add the restriction that only a finite number of elements of the sequence are different from zero. Now our ordering works, and it looks like the ordering of natural numbers written in decimal notation, except with digit positions reversed, and with arbitrary natural numbers instead of just the digits 0-9:

(0,0,0,...) < (1,0,0,0,...) < (2,0,0,0,...) < ... <

(0,1,0,0,0,...) < (1,1,0,0,0,...) < (2,1,0,0,0,...) < ... <

(0,2,0,0,0,...) < (1,2,0,0,0,...) < (2,2,0,0,0,...)

< ... <

(0,0,1,0,0,0,...) < (1,0,1,0,0,0,...) < (2,0,1,0,0,0,...)

< ...

So in general, to raise a well ordered set B to the power of another well ordered set E, we write down copies of the well ordered set E, and then map each element to some element of B, with the restriction that all but a finite number of elements of the domain E must map to the least element of the range B. That mapping is then an element of the well ordered set which is the power B^E. We find $1^{\omega }=1$, $2^{\omega }=\omega$, $2^{\omega +1}=\omega 2=\omega +\omega$.

The order type of the power B^E is the ordinal which results from applying ordinal exponentiation to the order type of the base B and the order type of the exponent E.

The definition of exponentiation can also be given inductively (the following induction is on β):

• α⁰ = 1,
• α^β+1 = (α^β)·α,
• and if δ is limit then α^δ is the limit of the α^β for all β<δ.

The properties of exponentiation are:

α⁰ = 1.

If 0 < α, then 0^α = 0.

1^α = 1.

α¹ = α.

Exponentiation is strictly increasing and continuous in the right argument:

γ > 1 and α < β $\iff$ γ^α < γ^β.

α ≤ β $\Longrightarrow$ α^γ ≤ β^γ.

However, 2 < 3 and yet 2^ω = 3^ω = ω.

Instead of a distributive law, we have α^β·α^γ = α^{β + γ}.

Instead of an associative law, we have (α^β)^γ = α^β·γ.

Cancellation: If α > 1 and α^β = α^γ, then β = γ. Also for all α and β, if 1 < β ≤ α, then there exist unique γ, δ and ρ such that α = β^γ · δ + ρ and 0 < δ < β and ρ < β^γ.

Warning: Ordinal exponentiation is quite different from cardinal exponentiation. For example, the ordinal exponentiation 2^ω = ω, but the cardinal exponentiation $2^{\aleph _{0}}$ = the cardinality of the continuum which is much larger than $\aleph _{0}$. To avoid confusing ordinal exponentiation with cardinal exponentiation, one uses symbols for ordinals in the former and symbols for cardinals in the latter.

Natural operations

The natural sum and natural product operations on ordinals were defined in 1906 by Hessenberg, and are sometimes called the Hessenberg sum (or product). (Jacobsthal defined a natural power operation in 1907, which is very rarely used.) They are also sometimes called the Conway operations, as Conway showed how to extend them to all surreal numbers. They have the advantage that they are associative and commutative, and natural product distributes over natural sum. The cost of making these operations commutative is that they lose the continuity in the right argument which is a property of the ordinary sum and product. The natural sum of α and β is sometimes denoted by α#β, and the natural product by a sort of doubled × sign. To define the natural sum of two ordinals, consider once again the disjoint union S$\cup$T of two well-ordered sets having these order types. Start by putting a partial order on this disjoint union by taking the orders on S and T separately but imposing no relation between S and T. Now consider the order types of all well-orders which extend this partial order: the least upper bound of all these ordinals (which is, actually, not merely a least upper bound but actually a greatest element) is the natural sum. Also, inductively, we can define the natural sum of α and β (by simultaneous induction on α and β) as the smallest ordinal greater than the natural sum of α and γ for all γ<β and of γ and β for all γ<α.

The natural sum is associative and commutative: it is always greater or equal to the usual sum, but it may be greater. For example, the natural sum of ω and 1 is ω+1 (the usual sum), but this is also the natural sum of 1 and ω.

To define the natural product of two ordinals, consider once again the cartesian product S×T of two well-ordered sets having these order types. Start by putting a partial order on this cartesian product by using just the product order (compare two pairs iff each of the two coordinates is comparable). Now consider the order types of all well-orders which extend this partial order: the least upper bound of all these ordinals (which is, actually, not merely a least upper bound but actually a greatest element) is the natural product. There is also an inductive definition of the natural product (by mutual induction), but it is somewhat tedious to write down and we will not do so (see the article on surreal numbers for the definition in that context, which, however, uses Conway subtraction, something which obviously cannot be defined on ordinals).

The natural product is associative and commutative and distributes over the natural sum: it is always greater or equal to the usual product, but it may be greater. For example, the natural product of ω and 2 is ω2 (the usual product), but this is also the natural product of 2 and ω.

Yet another way to define the natural sum and product is to use the Cantor normal form, taking into account the fact that, in the expression of the Cantor normal form, sums and products can also be considered as natural sums and natural products.

Cantor normal form

Ordinal numbers present a rich arithmetic. Every ordinal number α can be uniquely written as $\omega ^{\beta _{1}}c_{1}+\omega ^{\beta _{2}}c_{2}+\cdots +\omega ^{\beta _{k}}c_{k}$, where k is a natural number, $c_{1},c_{2},\ldots ,c_{k}$ are positive integers, and $\beta _{1}>\beta _{2}>\ldots >\beta _{k}$ are ordinal numbers (we allow $\beta _{k}=0$). This decomposition of α is called the Cantor normal form of α, and can be considered the positional base-ω numeral system. The highest exponent $\beta _{1}$ is called the degree of $\alpha$, and satisfies $\beta _{1}\leq \alpha$ (with equality if and only if $\alpha =\omega ^{\alpha }$, which can happen as explained below).

A minor variation of Cantor normal form, which is usually slightly easier to work with, is to set all the numbers c_i equal to 1 and allow the exponents to be equal. In other words, every ordinal number α can be uniquely written as $\omega ^{\beta _{1}}+\omega ^{\beta _{2}}+\cdots +\omega ^{\beta _{k}}$, where k is a natural number, and $\beta _{1}\geq \beta _{2}\geq \ldots \geq \beta _{k}\geq 0$ are ordinal numbers.

The Cantor normal form allows us to uniquely express—and order—the ordinals α which are built from the natural numbers by a finite number of arithmetical operations of addition, multiplication and “raising ω to the power of”: in other words, assuming $\beta _{1}<\alpha$ in the Cantor normal form, we can also express the exponents $\beta _{i}$ in Cantor normal form, and making the same assumption for the $\beta _{i}$ as for α and so on recursively, we get a system of notation for these ordinals (for example, $\omega ^{\omega ^{\omega 6+42}\cdot 1729+\omega ^{9}+88}+\omega ^{\omega ^{\omega }}+65537$ is one). It turns out that the ordinals α so defined are exactly the ordinals smaller than ε₀ (this can be taken as a definition of ε₀), where ε₀ is the smallest ordinal such that $\varepsilon _{0}=\omega ^{\varepsilon _{0}}$ (in other words, such that Cantor normal form does not produce exponents smaller than the ordinal one is trying to express). Or, if one wishes, ε₀ is exactly the set of finite arithmetical expressions of this form. It can also be seen as the limit of the sequence $\omega$, $\omega ^{\omega }$, $\omega ^{\omega ^{\omega }}$, etc. The ordinal ε₀ is important for various reasons in arithmetic (essentially because it measures the proof-theoretic strength of the first-order Peano arithmetic: that is, Peano's axioms can show transfinite induction up to any ordinal less than ε₀ but not up to ε₀ itself).

The Cantor normal form also allows us to compute sums and products of ordinals: to compute the sum, for example, one needs merely know that $\omega ^{\beta }c+\omega ^{\beta '}c'$ is $\omega ^{\beta '}c'$ if $\beta '>\beta$ (if $\beta '=\beta$ one can obviously rewrite this as $\omega ^{\beta }(c+c')$, and if $\beta '<\beta$ the expression is already in Cantor normal form); and to compute products, the essential fact is that when $\alpha =\omega ^{\beta _{1}}c_{1}+\cdots +\omega ^{\beta _{k}}c_{k}$ is in Cantor normal form (and α>0) then $\alpha \omega =\omega ^{\beta _{1}+1}$. Also $\alpha n=\omega ^{\beta _{1}}c_{1}n+\omega ^{\beta _{2}}c_{2}+\cdots +\omega ^{\beta _{k}}c_{k}$.

To compare two ordinals written in Cantor normal form, first compare $\beta _{1}$, then $c_{1}$, then $\beta _{2}$, then $c_{2}$, etc.. At the first difference, the ordinal which has the larger component is the larger ordinal. If one terminates before the other, then the one which terminates first is smaller.

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 a ordinal $\alpha$ is the smallest ordinal $\delta$ which is the order type of a cofinal subset of $\alpha$. The cofinality of a set of ordinals 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.

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. The ordinals 0, 1, $\omega$, and $\omega _{1}$ are regular, whereas 2, 3, and $\omega _{\omega }$ 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” ordinals

Generalities on recursive ordinals

Ordinal notations

Recursive ordinals (or computable ordinals) are certain countable ordinals: loosely speaking those which can be represented by a computable function. There are several equivalent definitions of this: the simplest is to say that a computable ordinal is the order-type of some recursive (i.e., computable) well-ordering of the natural numbers; so, essentially, an ordinal is recursive when we can present the set of smaller ordinals in such a way that a computer (Turing machine, say) can manipulate them (and, essentially, compare them).

A different definition uses Kleene's system of ordinal notations: briefly, an ordinal notation is either the name zero (describing the ordinal 0), or the successor of an ordinal notation (describing the successor of the ordinal described by that notation), or a Turing machine (computable function) which produces an increasing sequence or ordinal notations (describing the ordinal which is the limit of the sequence), and ordinal notations are (partially) ordered so as to make the successor of o greater than o and to make the limit greater than any term of the sequence (this order is computable; however, the set O of ordinal notations itself is highly non-recursive, owing to the impossibility of deciding whether a given Turing machine does indeed produce a sequence of notations); a recursive ordinal is then an ordinal which is described by some ordinal notation.

Any ordinal smaller than a recursive ordinal is itself recursive, so the set of all recursive ordinals forms a certain (countable) ordinal, the Church-Kleene ordinal (see below).

It is tempting to forget about ordinal notations and only speak of the recursive ordinals themselves: and some statements are made about recursive ordinals which, in fact, concern the notations for these ordinals. This can lead to difficulties, however, as even the smallest infinite ordinal, ω, has many notations, some of which cannot be proven to be equivalent to the obvious notation (the limit of the simplest program which enumerates all natural numbers).

Relation to systems of arithmetic

We briefly (and somewhat informally) mention an important relation between computable ordinals and certain formal systems (containing arithmetic, that is, at least a reasonable fragment of Peano arithmetic).

In a nutshell, the idea is that certain computable ordinals are so large that while they can be given by a certain ordinal notation o, a given formal system might not be sufficiently powerful to show that o is, indeed, an ordinal notation: the system does not show transfinite induction for such large ordinals.

For example, the usual first-order Peano axioms do not prove transfinite induction for (or beyond) ε₀: while the ordinal ε₀ can easily be arithmetically described (it is countable), the Peano axioms are not strong enough to show that it is indeed an ordinal; in fact, transfinite induction on ε₀ proves the consistency of Peano's axiom (a theorem by Gentzen), so by Gödel's incompleteness theorem, Peano's axiom cannot formalize that reasoning. (This is at the basis of the Kirby-Paris theorem on Goodstein sequences.) We say that ε₀ measures the proof-theoretic strength of Peano's axioms.

But we can do this for systems far beyond Peano's axioms. For example, the proof-theoretic strength of Kripke-Platek set theory is the Bachmann-Howard ordinal (see below), and, in fact, merely adding to Peano's axioms the axioms which state the well-ordering of all ordinals below the Bachmann-Howard ordinal is sufficient to obtain all arithmetical consequences of Kripke-Platek set theory.

Specific recursive ordinals

Predicative definitions and the Veblen hierarchy

We have already mentioned 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. The next ordinal satisfying this equation is called ε₁: it is the limit of the sequence

$\varepsilon _{0}+1$, $\omega ^{\varepsilon _{0}+1}=\varepsilon _{0}\cdot \omega$, $\omega ^{\omega ^{\varepsilon _{0}+1}}=(\varepsilon _{0})^{\omega }$, etc.

More generally, the $\iota$-th ordinal such that $\omega ^{\alpha }=\alpha$ is called $\varepsilon _{\iota }$. We could define $\eta _{0}$ as the smallest ordinal such that $\varepsilon _{\alpha }=\alpha$, but since the greek alphabet does not have transfinitely many letters it is better to use a more robust notation: define ordinals $\varphi _{\gamma }(\beta )$ by transfinite induction as follows: let $\varphi _{0}(\beta )=\omega ^{\beta }$ and let $\varphi _{\gamma +1}(\beta )$ be the $\beta$-th fixed point of $\varphi _{\gamma }$ (i.e., the $\beta$-th ordinal such that $\varphi _{\gamma }(\alpha )=\alpha$; so for example, $\varphi _{1}(\beta )=\varepsilon _{\beta }$), and when $\delta$ is a limit ordinal, define $\varphi _{\delta }(\alpha )$ as the $\alpha$-th common fixed point of the $\varphi _{\gamma }$ for all $\gamma <\delta$. This family of functions is known as the Veblen hierarchy. (There are inessential variations in the definition, such as letting, for $\delta$ a limit ordinal, $\varphi _{\delta }(\alpha )$ be the limit of the $\varphi _{\gamma }(\alpha )$ for $\gamma <\delta$: this essentially just shifts the indices by 1, which is harmless.)

Ordering: $\varphi _{\alpha }(\beta )<\varphi _{\gamma }(\delta )$ if and only if either ($\alpha =\gamma$ and $\beta <\delta$) or ($\alpha <\gamma$ and $\beta <\varphi _{\gamma }(\delta )$) or ($\alpha >\gamma$ and $\varphi _{\alpha }(\beta )<\delta$).

Fundamental sequences for the Veblen hierarchy

The fundamental sequence of an ordinal with cofinality ω is a distinguished strictly increasing ω-sequence which has the ordinal as its limit. Here we will describe fundamental sequences for the Veblen hierarchy of ordinals.

A variation of Cantor normal form used in connection with the Veblen hierarchy is -- every ordinal number α can be uniquely written as $\varphi _{\beta _{1}}(\gamma _{1})+\varphi _{\beta _{2}}(\gamma _{2})+\cdots +\varphi _{\beta _{k}}(\gamma _{k})$, where k is a natural number and each term after the first is less than or equal to the previous term and each $\gamma _{j}$ is not a fixed point of $\varphi _{\beta _{j}}$. If a fundamental sequence can be provided for the last term, then that term can be replaced by such a sequence to get a fundamental sequence for α.

No such sequence can be provided for $\varphi _{0}(0)$ = ω⁰ = 1 because it does not have cofinality ω.

For $\varphi _{0}(\gamma +1)$ = ω^{$\gamma +1$} = ω^$\gamma$·ω, we choose the function which maps the natural number m to ω^$\gamma$·m.

If γ is a limit which is not a fixed point of $\varphi _{0}$, then for $\varphi _{0}(\gamma )$, we replace γ by its fundamental sequence inside $\varphi _{0}$.

For $\varphi _{\beta +1}(0)$, we use 0, $\varphi _{\beta }(0)$, $\varphi _{\beta }(\varphi _{\beta }(0))$, $\varphi _{\beta }(\varphi _{\beta }(\varphi _{\beta }(0)))$, etc..

For $\varphi _{\beta +1}(\gamma +1)$, we use $\varphi _{\beta +1}(\gamma )+1$, $\varphi _{\beta }(\varphi _{\beta +1}(\gamma )+1)$, $\varphi _{\beta }(\varphi _{\beta }(\varphi _{\beta +1}(\gamma )+1))$, etc..

If γ is a limit which is not a fixed point of $\varphi _{\beta +1}$, then for $\varphi _{\beta +1}(\gamma )$, we replace γ by its fundamental sequence inside $\varphi _{\beta +1}$.

Now suppose that β is a limit: If $\beta <\varphi _{\beta }(0)$, then for $\varphi _{\beta }(0)$, we replace β by its fundamental sequence.

For $\varphi _{\beta }(\gamma +1)$, use $\varphi _{\beta _{m}}(\varphi _{\beta }(\gamma )+1)$ where $\beta _{m}$ is the fundamental sequence for β.

If γ is a limit which is not a fixed point of $\varphi _{\beta }$, then for $\varphi _{\beta }(\gamma )$, we replace γ by its fundamental sequence inside $\varphi _{\beta }$.

Otherwise, the ordinal cannot be described in terms of smaller ordinals using $\varphi$ and this scheme does not apply to it.

The Feferman-Schütte ordinal and beyond

The smallest ordinal such that $\varphi _{\alpha }(0)=\alpha$ is known as the Feferman-Schütte ordinal and generally written $\Gamma _{0}$. It can be described as the set of all ordinals which can be written as finite expressions, starting from zero, using only the Veblen hierarchy and addition. The Feferman-Schütte ordinal is important because, in a sense that is rather complicated to make precise, it is the smallest (infinite) ordinal which cannot be (“predicatively”) described using smaller ordinals. It measures the strength of such systems as “arithmetical transfinite recursion”.

More generally, Γ_α enumerates the ordinals that cannot be obtained from smaller ordinals using addition and the Veblen functions.

It is, of course, possible to describe ordinals beyond the Feferman-Schütte ordinal. One could continue to seek fixed points in more and more complicated manner: enumerate the fixed points of $\alpha \mapsto \Gamma _{\alpha }$, then enumerate the fixed points of that, and so on, and then look for the first ordinal α such that α is obtained in α steps of this process, and continue diagonalizing in this ad hoc manner. This leads to the definition of the “small” and “large” Veblen ordinals ($\psi (\Omega ^{\Omega ^{\omega }})$ and $\psi (\Omega ^{\Omega ^{\Omega }})$ with the notation introduced below), but one must run out of patience before one runs out of ordinals (even recursive - and therefore countable! - ones).

Impredicative ordinals

To go far beyond the Feferman-Schütte ordinal, one needs to introduce new methods. Unfortunately there is not yet any standard way to do this: every author in the subject seems to have invented their own system of notation, and it is quite hard to translate between the different systems. The first such system was introduced by Bachmann in 1950 (in an ad hoc manner), and different extensions and variations of it were described by Buchholz, Takeuti (ordinal diagrams), Feferman (θ systems), Aczel, Bridge, Schütte, and Pohlers. However most systems use the same basic idea, of constructing new countable ordinals by using the existence of certain uncountable ordinals. Here is perhaps the simplest way of doing this as an example:

• ψ(α) is defined to be the smallest ordinal that cannot be constructed by starting with 0 and Ω, and repeatedly applying addition, the Veblen functions, and ψ to previously constructed ordinals (except that ψ can only be applied to arguments less than α, to ensure that it is well defined).

Here Ω is the first uncountable ordinal. It is put in because otherwise the function ψ gets "stuck" at the smallest ordinal σ such that Γ_σ=σ: in particular ψ(α)=σ for any ordinal α satisfying σ≤α≤Ω. However the fact that we included Ω allows us to get past this point: ψ(Ω+1) is greater than σ. The key property of Ω that we used is that it is greater than σ. This definition is impredicative, because it uses the uncountable ordinal Ω, which in some sense already uses all the countable ordinals we are trying to construct in its construction.

To construct still larger ordinals, we can extend the definition of ψ by throwing in more ways of constructing uncountable ordinals. There are several ways to do this:

• We can add in more functions producing cardinals from ordinals such as the the $\omega$ function: whenever we have an ordinal α we add in the ordinal (in fact cardinal) $\omega _{\alpha }$.
• We can add in some inaccessible cardinals, or functions producing inaccessible cardinals.
• The function ψ defined above only constructs countable ordinals. We can modify it in various ways so that it also produces new uncountable ordinals (which can then be fed back into it to produce new countable ordinals). For example, instead of taking the smallest ordinal we have not yet produced as its value, we could take the smallest new ordinal of some given cardinality.

The Bachmann-Howard ordinal (ψ(ε_Ω+1) with the notation above) is an important one, because it describes the proof-theoretic strength of Kripke-Platek set theory. Indeed, the main importance of these large ordinals, and the reason to describe them, is their relation to certain formal systems as explained above. However, such powerful formal systems as full second-order arithmetic, let alone Zermelo-Fraenkel set theory, seem beyond reach for the moment.

By dropping the requirement of having a useful description, even larger recursive countable ordinals can be obtained as the ordinals measuring the strengths of various strong theories; roughly speaking, these ordinals are the smallest ordinals that the theories cannot prove are well ordered. By taking stronger and stronger theories such as second-order arithmetic, Zermelo set theory, Zermelo-Fraenkel set theory, or Zermelo-Fraenkel set theory with various large cardinal axioms, one gets some extemely large recursive ordinals. (Strictly speaking it is not known that all of these really are ordinals: by construction, the ordinal strength of a theory can only be proven to be an ordinal from an even stronger theory. So for the large cardinal axioms this becomes quite unclear.)

Beyond recursive ordinals

An even larger countable ordinal can be defined as the smallest countable ordinal which cannot be described in a recursive way (it is not the order type of any recursive well-ordering of the integers), and that ordinal is called the Church-Kleene ordinal, $\omega _{1}^{\mathrm {CK} }$ (despite the $\omega _{1}$ in the name, this ordinal is countable). Thus, $\omega _{1}^{\mathrm {CK} }$ is the smallest non-recursive ordinal, and there is no hope of precisely “describing” any ordinals from this point on—we can only define them. But it is still far from the first uncountable ordinal $\omega _{1}$ (although as its symbol suggests, it behaves in many ways rather like $\omega _{1}$).

The Church-Kleene ordinal is again related to Kripke-Platek set theory, but now in a different way: whereas the Bachmann-Howard ordinal was the smallest ordinal for which KP does not prove transfinite induction, the Church-Kleene ordinal is the smallest α such that the construction of the Gödel universe, L, up to stage α, yields a model $L_{\alpha }$ of KP: such ordinals are called admissible, thus $\omega _{1}^{\mathrm {CK} }$ is the smallest admissible ordinal (beyond ω in case the axiom of infinity is not included in KP). By a theorem of Sacks, the countable admissible ordinals are exactly those which are constructed in a manner similar to the Church-Kleene ordinal but for Turing machines with oracles. One sometimes writes $\omega _{\alpha }^{\mathrm {CK} }$ for the $\alpha$-th ordinal which is either admissible or limit of admissible; an ordinal which is both is called recursively inaccessible: there exists a theory of large ordinals in this manner which is highly parallel to that of (small) large cardinals (we can define recursively Mahlo cardinals, for example). But note that we are still talking about countable ordinals here!

We can imagine even larger ordinals which are still countable: for example, if ZFC has a transitive model (a hypothesis stronger than the mere hypothesis of consistency, and which is implied by the existence of an inaccessible cardinal), then there exists a countable $\alpha$ such that $L_{\alpha }$ is a model of ZFC. Even larger countable ordinals can be defined by indescribability conditions.

For large ordinals which are no longer countable, see the articles on large cardinals.

References for large ordinals

Most books describing large ordinals are on proof theory, and unfortunately tend to be out of print.

• W. Pohlers, Proof theory, ISBN 0387518428 (for Veblen hierarchy and some impredicative ordinals). This is probably the most readable book on large ordinals (which is not saying much).
• G. Takeuti, Proof theory, 2nd edition 1987 ISBN 0444104925 (for ordinal diagrams)
• K. Schütte, Proof theory, Springer 1977 ISBN 0387079114 (for Veblen hierarchy and some impredicative ordinals)
• Smorynski, C. The varieties of arboreal experience Math. Intelligencer 4 (1982), no. 4, 182-189; contains an informal description of the Veblen hierarchy.
• Theory of Recursive Functions and Effective Computability by Hartley Rogers ISBN 0262680521 (describes recursive ordinals and the Church–Kleene ordinal)
• Larry W. Miller,Normal Functions and Constructive Ordinal Notations,The Journal of Symbolic Logic,volume 41,number 2,June 1976,pages 439 to 459.

Topology and ordinals

Ordinals as topological spaces

Any ordinal can be made into a topological space by endowing it with the order topology (since, being well-ordered, an ordinal is in particular totally ordered): 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. (Note that if we are willing to accept a proper class as a topological space, then the class of all ordinals is also a topological space for the order topology.)

The set of limit points of an ordinal α is precisely the set of limit ordinals less than α. Successor ordinals (and zero) less than α are isolated points in α. In particular, the finite ordinals and ω are discrete topological spaces, and no ordinal beyond that is discrete. The ordinal α is compact as a topological space if and only if α is a successor ordinal.

The closed sets of a limit ordinal α 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, α+1 is obtained by taking the one-point compactification of α (if α is a limit ordinal; if it is not, α+1 is merely the disjoint union of α and a point), and for δ a limit ordinal, δ is equipped with the inductive limit topology.

As topological spaces, all the ordinals are Hausdorff and even normal. They are also totally disconnected (connected components are points), scattered (=every non-empty set has an isolated point; in this case, just take the smallest element), zero-dimensional (=the topology has a clopen basis: here, write an open interval (β,γ) as the union of the clopen intervals (β,γ'+1)=[β+1,γ'] for γ'<γ). However, they are not extremally disconnected in general (there is an open set, namely ω, whose closure is not open).

The topological spaces ω₁ and its successor ω₁+1 are frequently used as text-book examples of non-countable topological spaces. For example, in the topological space ω₁+1, the element ω₁ is in the closure of the subset ω₁ even though no sequence of elements in ω₁ has the element ω₁ as its limit. The space ω₁ is first-countable, but not second-countable, and ω₁+1 has neither of these two properties, despite being compact. It is also worthy of note that any continuous function from ω₁ to R (the real line) is eventually constant: so the Stone-Čech compactification of ω₁ is ω₁+1, just as its one-point compactification (in sharp contrast to ω, whose Stone-Čech compactification is much larger than ω₁).

Ordinal-indexed sequences

If α is a limit ordinal and X is a set, an α-indexed sequence of elements of X merely means a function from α to X. If X is a topological space, we say that an α-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 β<α such that x_ι is in U for all ι≥β. 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 (ω-indexed) sequences to determine limits in topology: for example, ω₁ is a limit point of ω₁+1 (because it is a limit ordinal), and, indeed, it is the limit of the ω₁-indexed sequence which maps any ordinal less than ω₁ to itself: however, it is not the limit of any ordinary (ω-indexed) sequence in ω₁, since any function from the natural numbers to ω₁ is bounded. However, ordinal-indexed sequences are not powerful enough to replace nets (or filters) in general: for example, on the Tychonoff plank (the product space $(\omega _{1}+1)\times (\omega +1)$), the corner point $(\omega _{1},\omega )$ is a limit point (it is in the closure) of the open subset $\omega _{1}\times \omega$, but it is not the limit of an ordinal-indexed sequence.

See also

• successor ordinal
• limit ordinal
• cardinal number
• serial number
• nominal 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. (1965). Cardinal and Ordinal Numbers (2nd ed.). Warszawa: Pantswowe Wydawnictwo Naukowe. Also defines ordinal operations in terms of the Cantor Normal Form.