Cardinal and Ordinal Numbers: An Informal Introduction

Cardinal numbers measure the size of a set, including infinite sets, while ordinal numbers describe the structure of a well-ordered set. Every set can be assigned a cardinal number by first well-ordering it (a theorem due to Zermelo) and then taking the smallest ordinal associated with that well-ordering. The smallest infinite cardinal is written $\aleph_0$ (aleph nought), and it behaves very differently from finite numbers under addition and multiplication.

Key Takeaways

• The cardinality of a finite set like {1,2,4,5} is simply the count of its elements, written |{1,2,4,5}|=4.
• For disjoint finite sets A and B, cardinality is additive: |A|+|B|=|A∪B|.
• The cardinality of a cartesian product equals the product of the cardinalities: |A×B|=|A|·|B|.
• Countably infinite sets are assigned the cardinal $\aleph_0$, and both $\aleph_0+\aleph_0$ and $\aleph_0\cdot \aleph_0$ equal $\aleph_0$ itself.
• An ordinal number is defined as a well-ordered set in which every element equals its own segment, a recursive definition built starting from the empty set.
• The continuum hypothesis asks whether any set exists with cardinality strictly between that of the natural numbers and the real numbers, and this question cannot be resolved within standard set theory.

How Do Cardinal Numbers Work for Finite Sets?

For finite sets, cardinal numbers are exactly the natural numbers familiar from arithmetic. The set {1,2,4,5} has cardinality 4, because it contains four elements. This is written |{1,2,4,5}|=4.

The operations on natural numbers correspond directly to operations on finite sets. Take two disjoint sets, such as {1,2,5} and {3,6,7,8}. Their union is {1,2,3,5,6,7,8}, and the cardinality of that union is the sum of the cardinalities of the two original sets:

$$|\{1,2,5\}|+|\{3,6,7,8\}|=|\{1,2,3,5,6,7,8\}|$$

This holds for any disjoint finite sets A and B:

$$|A|+|B|=|A\cup B|$$

The cartesian product of two sets is the set of all ordered pairs drawn from them:

$$A\times B=\{(a,b)~\vert~a\in A,b\in B\}$$

For example, $\{1,2\}\times \{1,2,3\}=\{(1,1),(1,2),(1,3),(2,1),(2,2),(2,3)\}$. The cardinality of a cartesian product is the product of the cardinalities of the two sets involved:

$$|A\times B|=|A|\cdot |B|$$

How Do Cardinal Numbers Extend to Infinite Sets?

Countable sets are the smallest infinite sets, and they are assigned the cardinal number $\aleph_0$ (read “aleph nought”; aleph is the first letter of the Hebrew alphabet). Operations on this cardinal are defined using the same logic applied to finite sets.

Since the union of two countable sets is itself countable, adding two copies of $\aleph_0$ produces $\aleph_0$ again:

$$\aleph_0+\aleph_0=\aleph_0$$

Similarly, adding any finite number n to $\aleph_0$ leaves it unchanged:

$$n+\aleph_0=\aleph_0$$

The product $\aleph_0\cdot \aleph_0$ is defined as the cardinality of the product of two countable sets, which is itself countable. So:

$$\aleph_0\cdot \aleph_0=\aleph_0$$

Cardinal arithmetic does not behave like finite arithmetic once infinite sets are involved. Assigning cardinal numbers to larger infinite sets requires first understanding ordinal numbers, which describe the internal structure of well-ordered sets.

What Is a Well-Ordered Set?

An ordered set is a set X equipped with a relation $\leq$ satisfying three conditions: every element relates to itself ($x\leq x$); if $x\leq y$ and $y\leq x$ then x=y; and if $x\leq y$ and $y\leq z$ then $x\leq z$.

Many familiar sets carry a natural order. The natural numbers $\mathbb{N}$ are ordered as:

$$0<1<2<3<4<5<…$$

The real numbers $\mathbb{R}$ are also ordered, but not well-ordered.

A well-ordered set is one in which every non-empty subset has a minimum element. $\mathbb{N}$ is well-ordered: the subset {1,2,4,5,7} has minimum 1, and the entire set has minimum 0. The real numbers are not well-ordered, since the open interval ]0,1[ has no minimum element, because 0 itself does not belong to that interval.

Every well-ordered set X has a smallest element $x_0$, a second-smallest element $x_1$ found by taking the minimum of $X\setminus \{x_0\}$, and so on. Adjoining a single new element $\omega$ to $\mathbb{N}$, with the rule that $n<\omega$ for every natural number n, produces another well-ordered set. Unlike $\mathbb{N}$, this new set has a maximum element.

How Are Ordinal Numbers Defined?

Ordinal numbers assign to every well-ordered set a value that fully describes its structure. The finite well-ordered set $0<1<2$ is assigned the ordinal number 3, since it contains three elements. Any other three-element well-ordered set, such as $1<4<1000$, is assigned the same ordinal number 3, because the two sets share the same structure.

The segment of an element a in a well-ordered set X is defined as every element strictly smaller than a:

$$X_a=\{x\in X~\vert~x<a\}$$

In $\mathbb{N}$, the segment of 6 is {0,1,2,3,4,5}.

An ordinal number is a well-ordered set in which every element equals its own segment, meaning $a=X_a$ for every a in X. This recursive definition starts with the empty set, which is vacuously an ordinal. The set $\{\emptyset\}$ is an ordinal, since the segment of its only element is empty, matching that element under the definition 0:=∅. The set $\{\emptyset,\{\emptyset\}\}$, ordered by $\emptyset<\{\emptyset\}$, is also an ordinal.

Using the notation 0:=∅, 1:={0}, 2:={0,1}, 3:={0,1,2}, and in general n:={0,1,…,n-1}, every natural number becomes an ordinal number. The set $\mathbb{N}$ itself is an ordinal, denoted $\omega$, since the segment of n is {0,1,…,n-1}, which equals n.

Ordinal construction continues indefinitely beyond $\omega$:

$\omega+1:=\omega\cup \{\omega\}$
$\omega+2:=\omega\cup \{\omega,\omega+1\}$
In general, $\omega+n:=\omega\cup \{\omega+1,…,\omega+n-1\}$

Further along, $\omega\cdot 2:=\{0,1,2,3,…,\omega,\omega+1,\omega+2,\omega+3,…\}$ is also an ordinal, and the process continues by defining $\omega\cdot 2+n=\omega\cdot 2 \cup\{\omega\cdot 2+1,…,\omega\cdot 2+n-1\}$. Every well-ordered set has the exact same structure as one of these ordinal numbers, which is why ordinals describe well-ordered sets completely.

How Do Ordinal Numbers Define Cardinal Numbers?

A theorem due to Zermelo states that any set X can be well-ordered, though usually in many different ways. The natural numbers $\mathbb{N}$ admit several well-orderings, each producing a different ordinal:

Different well-orderings of the natural numbers and their associated ordinals

Well-ordering	Associated ordinal
0<1<2<3<… (the usual order)	$\omega$
1<0<2<3<…	$\omega$
1<2<3<4<…<0	$\omega+1$
4<5<6<7<…<0<1<2<3	$\omega+4$

The cardinality of a set X is the smallest ordinal that can be assigned to it among all its possible well-orderings. $\mathbb{N}$ has cardinality $\omega$, but to avoid confusion between the ordinal and cardinal roles, this smallest infinite cardinal is written $\aleph_0$ rather than $\omega$ when discussing cardinality specifically.

The cardinals therefore run 0,1,2,3,4,…,$\aleph_0$, and the smallest cardinal larger than $\aleph_0$ is called $\aleph_1$, followed by $\aleph_2$, and so on for every ordinal $\alpha$, giving a cardinal $\aleph_\alpha$. This sequence extends far enough to produce a cardinal such as $\aleph_\omega$.

The set of rational numbers $\mathbb{Q}$ is countable, so its cardinality is $\aleph_0$. The set of real numbers $\mathbb{R}$ has cardinality $\aleph_\alpha$ for some ordinal $\alpha$, but which ordinal remains unknown. The continuum hypothesis asks whether any infinite set C exists with cardinality strictly between that of $\mathbb{N}$ and $\mathbb{R}$:

$$|\mathbb{N}|<C<|\mathbb{R}|$$

This question has been shown to be independent of the standard axioms of set theory, meaning it cannot be proven true or false within that system. As a result, the exact cardinality of $\mathbb{R}$ remains unknown, even though it is known to have one.

Frequently Asked Questions

What is the difference between a cardinal number and an ordinal number?

A cardinal number measures the size of a set, while an ordinal number describes the structure of a well-ordered set. Every cardinal number is defined as the smallest ordinal that can be assigned to a set among all its possible well-orderings.

Why does $\aleph_0+\aleph_0$ equal $\aleph_0$ instead of a larger number?

Because the union of two countable sets is itself countable. Since $\aleph_0$ is defined as the cardinality of any countable set, and that union remains countable, its cardinality stays $\aleph_0$ rather than growing.

Is the set of real numbers well-ordered under its usual order?

No. The usual order on the real numbers is not a well-order, because some non-empty subsets have no minimum element. The open interval ]0,1[ is one example, since 0 does not belong to it and no smallest number greater than 0 exists.

What does it mean for an ordinal to equal its own segment?

It means that, for every element a in the well-ordered set, the set of elements smaller than a is itself equal to a under the recursive definition built from the empty set. This self-referential property is what formally defines an ordinal number.

Has the continuum hypothesis been solved?

No. It has been shown to be independent of the standard axioms of set theory, meaning it can neither be proven nor disproven within that framework. Consequently, whether a set exists with cardinality strictly between $\mathbb{N}$ and $\mathbb{R}$ remains an open question in a formal sense.

Where can I read more about this topic?

Introduction to Set Theory by Karel Hrbacek and Thomas Jech offers an elementary introduction to cardinal and ordinal numbers.

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