Natural numbers have two faces: on one hand, they can be seen as sequences or “enumerations”—what we call ordinal numbers. On the other hand, they are perceived as “quantities,” which leads us to cardinal numbers. While this distinction is not always crucial in classical arithmetic, it becomes critically important when dealing with infinite sets.
Surprisingly, in the realm of infinity, the same quantity can give rise to different enumerations. For a rigorous exploration of the world of infinite numbers, it becomes essential to separate the discussion of ordinals from that of cardinals.
In this article, we will first dive into the fascinating world of ordinals, which are of intrinsic interest and serve as the foundation for the modern definition of cardinal numbers.

1. Ordinals and Cardinals: Two Distinct Aspects of Number

1.1. Distinguishing Between Order and Quantity

There are two fundamentally different ways to consider natural numbers. They are used either to enumerate the objects of a finite set (emphasizing the order in which they are arranged) or to count the quantity of objects in such a set (emphasizing the “power” they represent). In the first case, we speak of ordinal numbers, and in the second, of cardinal numbers.

In natural language, these two notions are well-differentiated: we refer to ordinal numeral adjectives (first, second, third…) or cardinal numeral adjectives (one, two, three…). However, in mathematics, which is primarily concerned with the reality of numbers rather than the signs describing them, these two notions merge when dealing with finite numbers, i.e., natural numbers. This is because, regardless of the order in which the objects of a finite set are counted, the same result is obtained.

1.2. Two Types of Numbers for Infinity

However, the order associated with a specific enumeration contains additional information beyond the quantity of objects, as it is generally possible to arrange the elements of the same finite set in different orders. This possibility of counting, or enumerating, finite sets naturally extends to all sets, including infinite ones, through the axiom of choice in set theory.

But when enumerating an infinite set, different results can be obtained! This is why, among infinite numbers—also called “transfinite” numbers—it is necessary to distinguish between ordinal numbers and cardinal numbers, with the latter being (well-defined) special cases of the former. Here, we address the concept of ordinal numbers, finite or infinite, as conceived in set theory, avoiding formal logic and instead relying on class theory (see Russell’s Paradox and Class Theory).

2. Well-Ordered Sets

2.1. Linear Orders

What does it mean to “enumerate” infinite sets? The notion of ordinals itself clarifies this concept, but originally, in Cantor’s work, it involved ordering the elements of a set, finite or infinite, in a manner ensuring there is always a smallest element and that enumerations can always be compared.

Recall that a linear order on a set $E$ is a binary relation, often denoted $<$, possessing the usual properties of strict order:
i) For no element $x$ in $E$ is $x < x$ (anti-reflexivity)
ii) If $x, y, z$ are elements of $E$ such that $x < y$ and $y < z$, then $x < z$ (transitivity)
iii) For any two elements $x, y$ in $E$, either $x < y$, $x = y$, or $x > y$ (totality).

For example, the strict order on the set $\mathbb{N}$ of natural numbers or on the set $\mathbb{R}$ of real numbers are linear orders.

2.2. Well-Orders

However, there is a fundamental difference between the linear orders on $\mathbb{N}$ and on $\mathbb{R}$: in the former case, there is a smallest element, which is also true for any nonempty subset of $\mathbb{N}$, as can be demonstrated using Peano’s axioms. In the case of $\mathbb{R}$, this property fails: for example, the set $\mathbb{R}_+^*$ of strictly positive real numbers has no smallest element!

This distinctive property of $\mathbb{N}$ also holds for its initial segments, i.e., subsets of the form $[[0,n[[ = {i \in \mathbb{N} : i < n}$ for a natural number $n$ (understood that for $n = 0$, $[[0,0[[ = \emptyset$ is the empty set). This concept serves as a model for the notion of “enumeration”:

Definition 1
A well-order on a set $E$ is a linear order $<$ on $E$ such that every nonempty subset of $E$ has a smallest element under $<$.

Thus, in classical set theory, an enumeration of any set $E$, finite or infinite, consists of a way to order its elements into a well-order.

Example 1
On the Cartesian product $\mathbb{N}^2 = \mathbb{N} \times \mathbb{N}$, a well-order is defined by ordering the pairs $(n, m)$ of natural numbers according to the lexicographic order (“dictionary order”): we say that $(n, m) < (n’, m’)$ if $n < n’$, or if $n = n’$ and $m < m’$. In this way, any nonempty subset $S$ of $\mathbb{N}^2$ has a smallest element: it suffices to choose the smallest integer $n$ appearing in a pair of the form $(n, m) \in S$, then choose the smallest integer $m$ appearing in such a pair for this choice of $n$. In lexicographic order, pairs of natural numbers are ordered as follows:
$$(0, 0), (0, 1), (0, 2), \ldots, (0, n), \ldots, (1, 0), (1, 1), (1, 2), \ldots, (1, n), \ldots, (2, 0), (2, 1), (2, 2), \ldots, (2, n), \ldots, (m, n), \ldots.$$

2.3. The Existence of an Enumeration

Is such an enumeration always possible? This is a theorem in set theory, which is actually equivalent to the axiom of choice:

(Zermelo’s) Theorem 1
On any set $E$, there exists a well-order relation.

In other words, it is always possible to “arrange” the elements of any set into an enumeration. The initial segments $[[0, n[[ $ of $\mathbb{N}$ are naturally arranged by strict order (including the empty set, which must be confirmed using the definition of a well-order!), but the set $\mathbb{N}$ itself is also ordered this way.

Now, if we consider each set $[[0, n[[ $ as an “ordinal” version of the number $n$ (it contains $n$ elements!), by analogy, the set $\mathbb{N}$ is composed of all “ordinal” numbers strictly less than “infinity.” In other words, the set $\mathbb{N}$ itself appears as a kind of infinite (ordinal) number, which we will now define precisely and rigorously.

3. Von Neumann Ordinals

3.1. Ordinals as Types of Well-Orders

Just as two sets have “the same number of elements” when there exists a bijection between them (and this number is called their “cardinal”), two well-ordered sets $(E_1, <_1)$ and $(E_2, <_2)$ have “the same type of order” when there exists a strictly increasing bijection between them. This means a bijection $f: E_1 \cong E_2$ such that for all $x, y \in E_1$, we have $x < y$ if and only if $f(x) < f(y)$.

Two well-ordered sets with the same type of order are indeed indistinguishable and should be identified, i.e., correspond to the same ordinal number. The initial definition of an ordinal, therefore, is that of a “type of well-order” in the following sense:

Definition 2
An ordinal number is a nonempty class $O$ of well-ordered sets, all having the same type of order, such that any well-ordered set with the same type of order as an element of $O$ is also in $O$.

In other words, Cantor’s original idea was to define an ordinal as a “type of well-order” through the class of all well-orders that can represent it, independent of the chosen representation.

Example 2
As in Example 1, the elements of the set $\mathbb{N} \times \{0,1\}$ can be ordered lexicographically by declaring $0 < 1$ on the second component. This means the pairs are ordered as follows:
$$(0,0), (0,1), (1,0), (1,1), (2,0), (2,1), \ldots.$$
It is easily shown that this is the same type of order as the natural linear order on $\mathbb{N}$: they both define the same ordinal. However, one can also define a well-order on $\mathbb{N} \times \{0,1\}$ by first listing all pairs of the form $(n, 0)$ and then all pairs of the form $(n, 1)$, i.e.,
$$(0,0), (1,0), (2,0), \ldots, (n,0), \ldots, (0,1), (1,1), (2,1), \ldots, (n,1), \ldots$$
The resulting well-order is equivalent to “two copies of $\mathbb{N}$ placed end-to-end.” In this case, every element of the form $(n, 0)$ is strictly less than $(0,1)$, which appears as an “infinite” element. This phenomenon does not occur with natural numbers, so this type of order is not that of $\mathbb{N}$ and defines a different ordinal number!

3.2. Von Neumann Ordinals

This equivalence-based approach works well for defining cardinal numbers (essentially the approach Gottlob Frege adopted in The Foundations of Arithmetic). However, it does not provide an ordinal as a “number” in the sense of a manipulable mathematical object, unless these classes are actual sets, which they are not in general.

John von Neumann proposed a description of ordinal numbers not as “classes of types of well-orders,” but as specific well-orders that represent each type and can thus be manipulated, particularly for calculations. Von Neumann ordinals are certain sets that generalize natural numbers in set theory and serve to count all sets, finite or infinite, in a standard way.

Since they represent ordinals in axiomatic set theory, von Neumann ordinals tend to replace Cantorian ordinals and are often referred to simply as “ordinals.”

3.3. Natural Numbers as Ordinals

This description of ordinals as sets is based on an idea of remarkable simplicity, involving two key concepts:
i) Building on the ordinal interpretation of each natural number $n$ as the set of natural numbers $i < n$, i.e., the initial segment $[[0, n[[ $;
ii) Generalizing the intuition that the set $\mathbb{N}$ itself is a kind of “infinite ordinal number,” composed of all those strictly less than it and obtained as a “limit” by their reunion.

Thus, the first ordinal number must represent the integer $0$, i.e., the empty set $\emptyset = [[0, 0[[$. Since we represent $0$ as $\mathbf{0} = \emptyset$, $1$ must be represented as $\mathbf{1} = \{\emptyset\}$, distinct from the previous one as it contains one element. Continuing, $\mathbf{2} = \{\mathbf{0}, \mathbf{1}\} = \{\emptyset, \{\emptyset\}\}$ which contains 2 elements, and so forth, each natural number $n$ is represented as the set $\mathbf{n}$ of ordinals strictly lesser than it. Moving from $n$ to $n+1$ is achieved via the “successor” operation defined for ordinals as $\mathbf{n+1} = s(\mathbf{n}) = \mathbf{n} \cup \{\mathbf{n}\}$.

From the empty set and elementary set operations (pairing and union), we can generate a representation of all natural numbers.

3.4. Ordinals as Sets

In this way, the set $\mathbb{N}$ can be represented as the class (in fact, the set, assuming $\mathbb{N}$ exists) denoted $\omega$ (the last letter of the Greek alphabet) of the representations $\mathbf{n}$ of natural numbers $n$. For any integer $n$, we have $\mathbf{n} = \{\mathbf{i} : i < n\}$.

However, $\omega$, so defined, has a peculiar property: it is both the union of the sets $\mathbf{n}$ and the set of all these sets! The sets of the form $\mathbf{n}$, as well as $\omega$, have a counterintuitive but ubiquitous property in set theory: each of their elements is also a subset of them. Such sets are called transitive.

This “transitivity” corresponds to the membership relation $\in$ between sets, which itself becomes the representation of the natural strict well-ordering relation among natural numbers! Through an extremely formal construction, we arrive at the rigorous definition of von Neumann ordinals:

Definition 3
A von Neumann ordinal is a transitive set $O$ for which the membership relation $\in$ is a well-order.

With this notion of ordinals, we finally have a proper mathematical object in the form of an actual set, manipulable within set theory, and capable of “counting” the elements of any set, finite or infinite. It can be shown that each “class of types of well-orders” (each Cantorian ordinal) contains exactly one von Neumann ordinal that represents it.

Yet, different results can still be obtained depending on how infinite sets are enumerated! Note that every element of an ordinal… is itself an ordinal: this follows directly from the definition.

4. The Burali-Forti Paradox

4.1. The Three Types of Ordinals

The Burali-Forti paradox is analogous to Russell’s paradox: there cannot exist a “set of all ordinals.” In other words, the class of ordinals is a proper class (a class that is not an element of any other class). This paradox can be demonstrated straightforwardly using the earlier framework: from now on, we will consider ordinals exclusively as von Neumann ordinals. Ordinals are typically denoted by lowercase Greek letters from the beginning of the alphabet: $\alpha, \beta, \gamma, \ldots$, with the last letter $\omega$ reserved to describe the ordinal number of the set $\mathbb{N}$ (with its natural strict order) and other infinite ordinals.

To demonstrate the paradox, some additional knowledge about ordinals is required, particularly that they can be classified into three distinct types. Recall that the successor of an ordinal $\alpha$ (already used to generate representations of natural numbers) is the set denoted $\alpha + 1 = \alpha \cup \{\alpha\}$. One must verify that this is indeed an ordinal. Additionally, recall that if $S$ is a set whose elements are themselves sets, then the union of $S$ is the set denoted $\bigcup S$, consisting of all objects $x$ that are elements of some element of $S$. If $s$ is a set of ordinals, the limit of $S$ is defined as its union, and it is easily verified to also be an ordinal. The trichotomy of ordinals can then be stated as follows:

Proposition 1
If $\alpha$ is an ordinal, then exactly one of the following three cases holds:
i) $\alpha = \mathbf{0}$ is the empty set;
ii) $\alpha$ is a successor ordinal (there exists an ordinal $\beta$ such that $\alpha = \beta + 1$);
iii) $\alpha$ is a nonzero limit ordinal ($\alpha \neq \mathbf{0}$ and $\alpha = \bigcup \alpha$ is the union of its elements, which are also its subsets).

Note that there are fundamentally only two types of ordinals: successor or limit (the null ordinal $\mathbf{0} = \emptyset$ being the union of the empty set, which is a set of ordinals, or even the union of the set $\{\emptyset\} = \mathbf{1}$).

4.2. The Class of All Ordinals

Consider the class denoted $Ord$ of all ordinals, which is permissible to form since von Neumann ordinals are sets. By definition of von Neumann ordinals, the meta-relation $\in$ of membership between sets induces on the class $Ord$ itself a strict order relation, which is actually a well-order (in the sense of classes)! Under this relation, every ordinal is the set of all ordinals strictly lesser than it. The following question then arises naturally: is the class $Ord$ itself an ordinal? It cannot be, as it would then be an element of itself. However, the only property it lacks is being a set, which leads to the content of the Burali-Forti paradox, stated as follows:

Theorem 2 (Burali-Forti Paradox)
The class $Ord$ of all ordinals is a proper class.

Let us explain this paradox through its proof: we proceed by contradiction, assuming that $Ord$ is a set, and then show that it must be an ordinal. In this case, an element of $Ord$ is an ordinal $\alpha$, and since an element of $\alpha$ is itself an ordinal, it is also an element of $Ord$, making $Ord$ a transitive set. Let $S$ be a nonempty subset of $Ord$: it is a set of ordinals, all of which are included in their union $\bigcup S$, which is itself a limit ordinal. Since $S$ is a nonempty subset of $\bigcup S$, it has a smallest element. Consequently, the set $Ord$ itself is by definition an ordinal and must therefore belong to itself: $Ord \in Ord$. This property is excluded by set theory (e.g., by the axiom of foundation), so by reductio ad absurdum, the class $Ord$ of all ordinals is not a set.

In summary, the class $Ord$ of all ordinals possesses all the properties of a von Neumann ordinal except for being a set: it is, in a sense, a “meta-ordinal,” the absolute analogue of the set $\omega$ of finite ordinals.

5. The Axiom of Infinity and Peano Arithmetic

The finite ordinals, i.e., the ordinals that are finite as sets, form a class denoted $\omega$. One must verify that this is the set of representations $\mathbf{n}$ of the natural numbers $n$ introduced earlier. Unlike the proper class $Ord$, the class $\omega$ is a set because of the axiom of infinity in set theory, which states that “an infinite ordinal exists.” Conversely, the axiom of infinity follows from the fact that the class $\omega$ of finite ordinals is a set, so these two properties are equivalent (within set theory minus the axiom of infinity), to the point where they are identified in natural set theory.

Indeed, the successor function, which associates to any ordinal $\alpha$ the ordinal $\alpha + 1$, acts on finite ordinals in the same way, and the basic properties of ordinals ensure that the following three axioms are satisfied:

• $\mathbf{0}$ is not the successor of any finite ordinal.
• If two finite ordinals have the same successor, then they are equal.
• If $C$ is a nonempty class of finite ordinals such that $\mathbf{0} \in C$ and $\mathbf{n+1} \in C$ whenever $\mathbf{n} \in C$, then $C$ is the class $\omega$.

These axioms are recognizable as a version of Peano arithmetic, which serves to axiomatize natural arithmetic and all of classical mathematics from set theory—a truly miraculous result. Thus, if the class $\omega$ is a set, Peano arithmetic has a realization (a “model”), which by definition is an infinite set. Conversely, if an infinite ordinal $O$ exists in the meta-universe, it can be shown that there is an injective function from $\omega$ into $O$, and hence that $\omega$ is a set.

In conclusion, the theory of numbers can be grounded in set theory, and all of mathematics can be reconstructed by assuming the “existence of infinity,” expressed by the following equivalence:

Theorem 3
The following properties are equivalent:
i) An infinite ordinal exists (axiom of infinity).
ii) The class $\omega$ of finite ordinals is a set.
iii) Peano arithmetic has a realization.

It is worth noting, however, that it can be shown without the axiom of infinity that the class $\omega$, and therefore also the class $Ord$, is infinite. But mathematics truly begins when an infinite class can be manipulated as an object—that is, as a set.