Ordinal number

This article is about the mathematical concept. For number words denoting a position in a sequence ("first", "second", "third", etc.), see Ordinal numeral.

Representation of the ordinal numbers up to $\omega ^{\omega }$ . One turn of the spiral corresponds to the mapping ⁠ $f(\alpha )=\omega (1+\alpha )$ ⁠. Since ![{\displaystyle f}](https://wikimedia.or…

This article is about the mathematical concept. For number words denoting a position in a sequence ("first", "second", "third", etc.), see Ordinal numeral.

Representation of the ordinal numbers up to $\omega ^{\omega }$ . One turn of the spiral corresponds to the mapping ⁠ $f(\alpha )=\omega (1+\alpha )$ ⁠. Since $f$ has $\omega ^{\omega }$ as the least fixed point, larger ordinal numbers cannot be represented on this diagram.

In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, nth, etc.) aimed to extend enumeration to infinite sets.[1]

A finite set can be enumerated by successively labeling each element with the least natural number that has not been previously used. To extend this process to various infinite sets, ordinal numbers are defined more generally using linearly ordered Greek letter variables that include the natural numbers and have the property that every set of ordinals has a least or "smallest" element (this is needed for giving a meaning to "the least unused element"). This more general definition allows us to define an ordinal number $\omega$ (omega) to be the least element that is greater than every natural number, along with ordinal numbers ⁠ $\omega +1$ ⁠, ⁠ $\omega +2$ ⁠, etc., which are even greater than ⁠ $\omega$ ⁠.

A linear order such that every non-empty subset has a least element is called a well-order. The axiom of choice implies that every set can be well-ordered, and given two well-ordered sets, one is isomorphic to an initial segment of the other. So ordinal numbers exist and are essentially unique.

Ordinal numbers are distinct from cardinal numbers, which measure the size of sets. Although the distinction between ordinals and cardinals is not this apparent on finite sets (one can go from one to the other just by counting labels), they are very different in the infinite case, where different infinite ordinals can correspond to sets having the same cardinal. Like other kinds of numbers, ordinals can be added, multiplied, and exponentiated, although none of these operations are commutative.

Ordinals were introduced by Georg Cantor in 1883[2] to accommodate infinite sequences and classify derived sets, which he had previously introduced in 1872 while studying the uniqueness of trigonometric series.[3]

Ordinals extend the natural numbers

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A natural number (which, in this context, includes the number 0) can be used for two purposes: to describe the size of a set, or to describe the position of an element in a sequence. When restricted to finite sets, these two concepts coincide, since all linear orders of a finite set are isomorphic.

However, for infinite sets, one has to distinguish between the notion of size, which leads to cardinal numbers, and the notion of position, which leads to the ordinal numbers described here. This is because while any set has only one size (its cardinality), there are many nonisomorphic well-orderings of any infinite set, as explained below.

Whereas the notion of cardinal number is associated with a set with no particular structure on it, the ordinals are intimately linked with the special kind of sets that are called well-ordered. A well-ordered set is a totally ordered set (an ordered set such that, given two distinct elements, one is less than the other) in which every non-empty subset has a least element. Equivalently, assuming the axiom of dependent choice, it is a totally ordered set without any infinite decreasing sequence — though there may be infinite increasing sequences. Ordinals may be used to label the elements of any given well-ordered set (the smallest element being labelled 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 that is not a label for an element of the set. This "length" is called the order type of the set.

Every ordinal is defined by the set of ordinals that precede it. The most common definition of ordinals identifies each ordinal as the set of ordinals that precede it. For example, the ordinal 42 is generally identified as the set {0, 1, 2, ..., 41}. Conversely, any set S of ordinals that is downward closed — meaning that for every ordinal α in S and every ordinal β < α, β is also in S — is (or can be identified with) an ordinal.

This definition of ordinals in terms of sets allows for infinite ordinals. The smallest infinite ordinal is ⁠ $\omega$ ⁠, which can be identified with the set of natural numbers (so that the ordinal associated with any natural number precedes $\omega$ ). 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 with it.

A graphical "matchstick" representation of the ordinal ω2. Each stick corresponds 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 formed in this way (the ω·m+n, where m and n are natural numbers) must itself have an ordinal associated with it, and that is ω2. Further on, there will be ω3, then ω4, and so on, and ωω, then ωωω, then later ωωωω, and even later ε0 (epsilon nought) (to give a few examples of relatively small—countable—ordinals). This can be continued indefinitely (as every time one says "and so on" when enumerating ordinals, it defines a larger ordinal). The smallest uncountable ordinal is the set of all countable ordinals, expressed as ω1 or ⁠ $\Omega$ ⁠.[4]

In a well-ordered set, every non-empty subset contains a distinct smallest element. Given the axiom of dependent choice, this is equivalent to saying that the set is totally ordered and there is no infinite decreasing sequence (the latter being 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 the computation will terminate.

It is inappropriate to distinguish between two well-ordered sets if they only differ in the "labeling of their elements", or more formally: if the elements of the first set can be paired off 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 (with the understanding that this is an equivalence relation).

Formally, if a partial order ≤ is defined on the set S, and a partial order ≤’ is defined on the set *S’ *, then the posets (S,≤) and (*S’ *,≤‘) are order isomorphic if there is a bijection f that preserves the ordering. That is, f(a) ≤’ f(b) if and only if a ≤ b. Provided there exists an order isomorphism between two well-ordered sets, the order isomorphism is unique: this makes it quite justifiable to consider the two 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. Every well-ordered set (S,<) is order-isomorphic to the set of ordinals less than one specific ordinal number under their natural ordering. This canonical set is the order type of (S,<).

Essentially, an ordinal is intended to be defined 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. The ordinal can be said to be the order type of any set in the class.

Definition of an ordinal as an equivalence class

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The original definition of ordinal numbers, found for example in the 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 axiomatic 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

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A von Neumann ordinal is defined as a set that represents a well-ordered class by being the set of all preceding ordinals. Formally, a set $S$ is an ordinal if and only if $S$ is transitive (every element of $S$ is a subset of $S$ ) and strictly well-ordered by set membership ( $\in$ ). Assuming the Axiom of regularity, a set is an ordinal if and only if it is transitive and strictly totally ordered by membership, as regularity prevents infinite descending chains. Without assuming regularity, the definition requires explicit well-ordering: membership must be trichotomous on $S$ and every non-empty subset must have a minimal element. These conditions imply that membership is irreflexive ( $x\notin x$ ) and transitive. The finite ordinals are defined inductively as $0=\emptyset$ , $1={0}$ , $2={0,1}$ , and so on.

0 = ∅ is an ordinal.
If α ∈ β and β is an ordinal, then α is an ordinal: transitivity is found by unpacking the definition of ordinal. In particular, transitivity of α as a set is found by finding that all the relevant ordinals are elements of β and then using the transitivity of the ∈ relation within β.[5]
If α ≠ β are both ordinals and α ⊂ β, then α ∈ β: let γ = min{β − α}. α is transitive so α = { ξ ∈ β | ξ ∈ γ } = γ ∈ β.[6] Note: this means that ⊆ is essentially the ≤ relation between ordinals.
If α and β are both ordinals, then α ⊆ β or β ⊆ α: γ = α ∩ β is an ordinal, so γ = α or γ = β, or else by (iii) γ ∈ γ, contradicting irreflexively.[6]
Any nonempty class of ordinals is a well-ordered class; if it is a set, then it is a well-ordered set.
If A is a nonempty set of ordinals, then $\min A=\bigcap A$ .[7]
If A is a set of ordinals, then $\sup A=\bigcup A$ is an ordinal.[7]
For any ordinal α, succ α = α ∪ {α} is an ordinal and succ α = min{β : β > α}.[7]
(Burali-Forti paradox) The class of all ordinals $ON$ is not a set, otherwise consider succ sup ON.[7]

Natural numbers and order types

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The set of natural numbers $\mathbb {N}$ is defined as the smallest inductive set (containing $\emptyset$ and closed under successor). It is an ordinal because induction demonstrates it is a transitive set composed of ordinals, which implies that the union of $\mathbb {N}$ is equal to $\mathbb {N}$ , and the above theorems imply that this implies that $\mathbb {N}$ is an ordinal. $\mathbb {N}$ is the least limit ordinal (see following section on limit and successor ordinals), meaning it is a non-zero ordinal with no immediate predecessor because induction also demonstrates that all elements of $\mathbb {N}$ are zero or a successor.

Every well-ordered set $S$ is order-isomorphic to exactly one ordinal, known as its order type. Uniqueness is guaranteed because a well-ordered set cannot be isomorphic to a proper initial segment of itself, preventing isomorphism to two distinct ordinals. The existence of this ordinal is proven by defining a map pairing each $x\in S$ with the ordinal representing the order type of the initial segment $S_{x}={y\in S\mid y<x}$ . By the Axiom schema of replacement, the range of this map is a set of ordinals. Because this range is downward closed (the order type of a segment of a segment is a smaller ordinal), the range is itself an ordinal $\gamma$ . The domain of the isomorphism must be all of $S$ ; otherwise, the least element $z\in S$ outside the domain would imply $S_{z}\cong \gamma$ , which would effectively include $z$ in the domain, a contradiction. Thus, $S\cong \gamma$ .

Successor and limit ordinals

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Every ordinal number is one of three types: the ordinal zero, a successor ordinal, or a limit ordinal.

There is variation in the definition of limit ordinals regarding the inclusion of zero. Some texts, such as Introduction to Cardinal Arithmetic by Holz et al., define a limit ordinal as a non-zero ordinal that is not a successor.[8] In contrast, other standard set theory texts, including Jech’s Set Theory and Just and Weese’s Discovering Modern Set Theory, define a limit ordinal simply as any ordinal that is not a successor, which implies that 0 is a limit ordinal.[9][10] When the topological definition is used (based on the order topology), 0 is not a limit ordinal because it is not a limit point of the set of smaller ordinals (which is empty); Rosenstein’s Linear Orderings uses this definition.[11] When 0 is included as a limit, ordinals that are strictly greater than 0 and not successors are usually referred to as "nonzero limit ordinals".

The following properties characterize nonzero limit ordinals:

$\lambda$ is a nonzero limit ordinal if and only if $\lambda \neq 0$ and for every ordinal $\alpha <\lambda$ , the successor $S(\alpha )$ is also less than $\lambda$ .

This implies that a nonzero limit ordinal is equal to the supremum of all ordinals strictly less than it:

$\lambda$ is a nonzero limit ordinal if and only if $\lambda =\sup{\alpha \mid \alpha <\lambda }=\bigcup \lambda$ and $\lambda \neq 0$ .

For example, $\omega$ is a limit ordinal because any natural number is less than $\omega$ , and the successor of any natural number is also a natural number (hence less than $\omega$ ).

Transfinite sequence

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If $\alpha$ is any ordinal and $X$ is a set, an $\alpha$ -indexed sequence of elements of $X$ is a function from $\alpha$ to $X$ . This concept, a transfinite sequence (if $\alpha$ is infinite) or ordinal-indexed sequence, is a generalization of the concept of a sequence. An ordinary sequence corresponds to the case $\alpha =\omega$ , while a finite $\alpha$ corresponds to a tuple, a.k.a. string.

While a sequence indexed by a specific ordinal $\alpha$ is a set, a sequence indexed by the class of all ordinals is a proper class. The Axiom schema of replacement guarantees that any initial segment of such a class-sequence (the restriction of the function to some specific ordinal $\delta$ ) is a set.

When $\langle x_{\iota }\mid \iota <\lambda \rangle$ is a transfinite sequence of ordinals indexed by a limit ordinal $\lambda$ and the sequence is increasing (i.e. $\iota <\rho \implies x_{\iota }<x_{\rho }$ ), its limit is defined as the least upper bound of the set ${x_{\iota }\mid \iota <\lambda }$ .

A transfinite sequence $f$ mapping ordinals to ordinals is said to be continuous (in the order topology) if for every limit ordinal $\lambda$ in its domain,

if f (λ) is a limit ordinal and for every ε < f (λ) there exists a δ < λ such that for every γ, if δ < γ < λ, then ε < f (γ) ≤ f (λ), and
if f (λ) is not a limit ordinal, there exists a δ < λ such that for every γ, if δ < γ < λ, then f (γ) = f (λ).

A sequence is called normal if it is both strictly increasing and continuous. If a sequence f is increasing (not necessarily strictly) and continuous and λ is a limit ordinal, then $f(\lambda )=\bigcup _{\beta <\lambda }f(\beta )$ .

Transfinite induction

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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 that 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

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Transfinite induction can be used not only to prove theorems but also to define functions on ordinals. This is known as transfinite recursion.

Formally, a function F is defined on the ordinals if, for every ordinal α, the value ⁠ $F(\alpha )$ ⁠ is specified using the set of values ${F(\beta )\mid \beta <\alpha }$ .

Very often, when defining a function F by transfinite recursion on all ordinals, the definition is separated into cases based on the type of the ordinal:

Base case: Define $F(0)$ .
Successor step: Define $F(\alpha +1)$ assuming $F(\alpha )$ is defined.
Limit step: For a limit ordinal $\lambda$ , define $F(\lambda )$ as the limit of $F(\beta )$ for all $\beta <\lambda$ (either in the sense of ordinal limits or some other notion of limit if the codomain allows it).

The interesting step in the definition is usually the successor step. If $F(\alpha )$ for limit ordinals $\alpha$ is defined as the limsup of $F(\beta )$ for $\beta <\alpha$ and $F$ takes ordinal values and is non-decreasing, the function $F$ will be continuous as defined above. Ordinal addition, multiplication and exponentiation are continuous as functions of their second argument.

The existence and uniqueness of such a function are proven by constructing it as the union of partial approximations. The proof proceeds in three steps:

Local Existence: For any specific ordinal δ, one proves the existence of a unique "recursion segment"—a function defined on δ that satisfies the recursive rule for all ⁠ $\beta <\delta$ ⁠.
Uniqueness and Compatibility: One proves that any two recursion segments agree on their common domain. If ⁠ $g_{1}$ ⁠ is a segment on ⁠ $\delta _{1}$ ⁠ and ⁠ $g_{2}$ ⁠ is a segment on ⁠ $\delta _{2}$ ⁠ with ⁠ $\delta _{1}<\delta _{2}$ ⁠, then ⁠ $g_{2}$ ⁠ restricted to ⁠ $\delta _{1}$ ⁠ is identical to ⁠ $g_{1}$ ⁠.
Global Definition: The global class function F is defined as the union of all such unique recursion segments. For any ordinal α, the value ⁠ $F(\alpha )$ ⁠ is the value assigned to α by any recursion segment defined on a domain larger than α.

The rigorous justification for local existence relies on the Axiom schema of replacement on the step of limit ordinals in order to collect the recursion segments into a set.

This construction allows definitions such as ordinal addition, multiplication, and exponentiation to be rigorous. For example, exponentiation ⁠ $\alpha ^{\beta }$ ⁠ is defined recursively on β:

The principle of transfinite recursion also implies that recursion can be performed up to a specific ordinal $\delta$ (defining a set rather than a proper class). This is formally achieved by defining a rule that returns a "throwaway" or default value (such as 0) if the recursion index exceeds $\delta$ . This technique is often used to define sequences of length $\omega$ . For example, to prove that a normal function $f$ has arbitrarily large fixed points, one constructs a sequence starting with any ordinal $\gamma _{0}$ and defines $\gamma _{n+1}=f(\gamma _{n})$ by recursion on $n<\omega$ . The limit $\delta =\sup _{n<\omega }\gamma _{n}$ is a fixed point of $f$ because continuity ensures $f(\delta )=f(\sup \gamma _{n})=\sup f(\gamma _{n})=\sup \gamma _{n+1}=\delta$ .

Indexing classes of ordinals

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Any well-ordered set is similar (order-isomorphic) to a unique ordinal number $\alpha$ ; in other words, 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 in the class of all ordinals, this puts it in class-bijection with the class of all ordinals). So the $\gamma$ -th element in the class (with the convention that the "0-th" is the smallest, the "1-st" is the next smallest, and so on) can be freely spoken of. 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 ⁠ $\beta <\gamma$ ⁠.

This could be applied, 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, one can consider additively indecomposable ordinals (meaning a nonzero ordinal that 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".

Arithmetic of ordinals

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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 that represents the operation or by using transfinite recursion. The Cantor normal form provides a standardized way of writing ordinals. It uniquely represents each ordinal as a finite sum of ordinal powers of ω. However, this cannot form the basis of a universal ordinal notation due to such self-referential representations as ε0 = ωε0.

Ordinals are a subclass of the class of surreal numbers, and the so-called "natural" arithmetical operations for surreal numbers are an alternative way to combine ordinals arithmetically. They retain commutativity at the expense of continuity.

Interpreted as nimbers, a game-theoretic variant of numbers, ordinals can also be combined via nimber arithmetic operations. These operations are commutative but the restriction to natural numbers is generally not the same as ordinary addition of natural numbers.

Ordinals and cardinals

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Initial ordinal of a cardinal

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Each ordinal associates with one cardinal, its cardinality. If there is a bijection between two ordinals (e.g. ω = 1 + ω and ω + 1 > ω), then they associate with the same cardinal. Any well-ordered set having an ordinal as its order-type has the same cardinality as that ordinal. The least ordinal associated with a given cardinal is called the initial ordinal of that cardinal. Every finite ordinal (natural number) is initial, and no other ordinal associates with its cardinal. But most infinite ordinals are not initial, as many infinite ordinals associate with the same cardinal. 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 theories with the axiom of choice, the cardinal number of any set has an initial ordinal, and one may employ the Von Neumann cardinal assignment as the cardinal’s representation. (However, we must then be careful to distinguish between cardinal arithmetic and ordinal arithmetic.) In set theories without the axiom of choice, a cardinal may be represented by the set of sets with that cardinality having minimal rank (see Scott’s trick).

One issue with Scott’s trick is that it identifies the cardinal number $0$ with ⁠ ${\emptyset }$ ⁠, which in some formulations is the ordinal number ⁠ $1$ ⁠. It may be clearer to apply Von Neumann cardinal assignment to finite cases and to use Scott’s trick for sets which are infinite or do not admit well orderings. Note that cardinal and ordinal arithmetic agree for finite numbers.

The α-th infinite initial ordinal is written ⁠ $\omega _{\alpha }$ ⁠, it is always a limit ordinal. 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 ω can be identified with ⁠ $\aleph _{0}$ ⁠, except that the notation $\aleph _{0}$ is used when writing cardinals, and ω when writing ordinals (this is important since, for example, $\aleph _{0}^{2}$ = $\aleph _{0}$ 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-ord