Cardinal number

Cardinal numbers describe the size of a collection or set, from simple counting numbers to infinite cardinalities such as aleph‑null and the continuum; central to counting and set theory.

Author: Leandro Alegsa Creation date: November 13, 2022 Last updated: May 28, 2026

Overview

A cardinal number names how many items are in a collection. In ordinary contexts these are the familiar counting words such as one, two, three and so on; they tell the size of a pile, group or list. In more formal terms the cardinal number associated with a collection is the abstract measure of its size without regard to order or labels. Everyday counting and much of elementary arithmetic rely on the concept of cardinal numbers; see cardinal numbers for a basic introduction.

Definition and basic properties

Given a set, its cardinal number (or its cardinality) is a way to classify the set by size. For finite sets this reduces to the ordinary natural number that counts elements. In modern mathematics, two sets are said to have the same cardinality if there is a one‑to‑one correspondence between their elements. That equivalence notion lets one compare sizes even when the sets are infinite: if such a bijection exists the sets are regarded as the same size.

Finite cardinals behave as expected: adding or removing elements changes the cardinal in the familiar way. Infinite cardinals, however, follow rules that often conflict with finite intuition and lead to surprising results.

Finite and infinite examples

The smallest infinite cardinal is the size of the set of all nonnegative integers. Georg Cantor introduced the notation aleph numbers to name infinite cardinals. The first of these is Aleph null (written ℵ0). $\aleph_0$ This cardinal counts sequences like 0, 1, 2, 3, ... and any set that can be put into a one‑to‑one correspondence with those integers is called countable. Examples of countable sets include the set of natural numbers $\aleph_0$ and the set of rational numbers $\aleph_0$ .

Other infinite cardinals are strictly larger. The set of real numbers has a different size from the integers; its size is often called the cardinality of the continuum and is commonly denoted by c. One may also compare c with aleph numbers: one convention labels certain infinite sizes ℵ1, ℵ2, and so on, and some of these play a role in foundational questions. $\aleph_1$ ${\mathfrak {c}}$

History and foundational results

The theory of infinite cardinals was developed in the late 19th century by Georg Cantor, who showed that infinite sets come in different sizes. His diagonal argument demonstrates that the real numbers cannot be matched one‑for‑one with the integers, so the continuum has strictly greater cardinality than ℵ0; this and related conclusions are sometimes summarized under Cantor's theorem. $\aleph_0$

Continuum hypothesis and significance

A central question raised by Cantor asks whether there exists a cardinal strictly between ℵ0 and the cardinality of the continuum. This statement is known as the continuum hypothesis. Modern set theory shows that its truth cannot be decided from the standard axioms of set theory (ZFC) alone, which places it among the most famous independent problems in mathematics.

Uses, distinctions and examples

Practical counting and combinatorics use finite cardinals to enumerate possibilities and outcomes.
In analysis and topology, the size of sets (finite, countable, or uncountable) often determines which constructions or theorems apply.
In logic and foundations, different infinite cardinals index hierarchies of infinite structures and influence independence results.
Careful distinction: ordinal numbers measure position or order type, while cardinals measure size.

Understanding cardinal numbers connects everyday counting to deep questions about infinity, structure and the limits of formal axioms. For introductory material consult standard texts and resources on cardinal numbers and set theory; for historical context see writings about Georg Cantor and his contributions.

Definition

Two sets and called equipotent if there is a bijection from to ; one then writes $\left\vert X\right\vert = \left\vert Y\right\vert$ or $X\sim Y$ . Equivalence $\sim$ is an equivalence relation on the class of all sets.

Cardinal numbers as real classes

The equivalence class of the set with respect to the relation of equality is called the cardinal number $\left\vert X\right\vert$ .

The problem with this definition is that the cardinals are then themselves not sets, but real classes. (With the exception of $\left\vert\emptyset\right\vert$ ).

This problem can be avoided by using $\left\vert X\right\vert$ not to denote the whole equivalence class, but to select an element from it, one selects a representative system, so to speak. In order to do this formally correct, one uses the theory of ordinal numbers, which must be defined accordingly beforehand in this approach:

Cardinal numbers as special ordinal numbers

Every set is equi-efficient to a well-ordered set (provided that one assumes the well-ordering theorem equivalent to the axiom of choice). To belongs an ordinal number. can be chosen such that this ordinal number becomes smallest possible, since ordinal numbers are themselves well-ordered; then is an initial number. One can equate the cardinal number $\left\vert A\right\vert$ with this smallest ordinal number.

By this set-theoretic handle, the cardinality of a set is itself again a set. It follows immediately the comparability theorem that the cardinals are totally ordered, because they are even well-ordered as a subset of the ordinals. This cannot be proved without the axiom of choice.

Motivation

Descriptively, cardinal numbers serve to compare the size of sets without having to refer to the appearance of their elements. For finite sets, this is easy. You simply count the number of elements. To compare the power of infinite sets, you need a little more work.

In the following, the terms at most equally powerful and less powerful are needed:

If there is a bijection from to a subset of then is called at most equal to . One then writes $\left\vert A\right\vert \leq \left\vert B\right\vert$ .

If there is a bijection from to a subset of but there isno bijection from to then said to be less powerful than and said to be more powerful than . One then writes $\left\vert A\right\vert < \left\vert B\right\vert$ .

These terms are explained in more detail in the article Thickness.

For example, for finite sets it holds that real subsets are less powerful than the whole set, whereas in Hilbert's Hotel article it is illustrated by an example that infinite sets have real subsets which are equally powerful to them.

In the study of these large sets, the question arises whether equally-powerful ordered sets necessarily have matching orders. It turns out that this is not so for infinite sets, e.g. the ordinary order of the natural numbers is $\N = \{0 < 1 < 2 < 3 < \dotsb\}$ different from the ordered set $A := \{0 < 1 < 2 < 3 < \dotsb < 0^\prime\}$ . The set is equi-empowered to $\mathbb {N}$ . Thus $f\colon 0\mapsto 1, 1\mapsto 2, 2\mapsto 3, \dots,0^\prime\mapsto 0$ a bijection, but in unlike $\mathbb {N}$ a largest element. Considering the order of sets, we arrive at ordinal numbers. The ordinal number of $\mathbb {N}$ is called ω $\omega$ and that of is ω $\omega+1$ .

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Creation date: November 13, 2022

Last updated: May 28, 2026