09-07-2026, 11:17 PM
Summary:
Cardinality is the mathematical notion used to describe the size of a set, including both finite and infinite sets. For a finite set $A$, its cardinality $|A|$ is simply the number of elements it contains. More generally, two sets $A$ and $B$ have the same cardinality if there exists a bijection between them—a one-to-one correspondence pairing every element of $A$ with exactly one element of $B$. This definition produces a striking feature of infinite sets: a proper subset can have the same cardinality as the whole set. For example, the natural numbers $\mathbb N$ and the even numbers have the same cardinality because $f(n)=2n$ gives a bijection between them.
Infinite sets are divided into countable and uncountable ones. The natural numbers, integers, and rational numbers are countably infinite and have cardinality $\aleph_0$. In contrast, Cantor's diagonal argument shows that the real numbers $\mathbb R$ are uncountable and therefore form a strictly larger infinity. More generally, Cantor's theorem says that for every set $A$, its power set $\mathcal P(A)$ has strictly greater cardinality than $A$, producing an endless hierarchy of increasingly large infinities. Cardinal numbers such as $\aleph_0,\aleph_1,\aleph_2,\ldots$ provide a systematic way of describing these different infinite sizes.
One of the central questions arising from this theory is the Continuum Hypothesis (CH): whether the cardinality of the real numbers is exactly the next cardinal after $\aleph_0$, that is, whether $|\mathbb R|=\aleph_1$. Remarkably, Gödel and Cohen's work ultimately established that CH can neither be proved nor disproved from the standard Zermelo–Fraenkel axioms with the Axiom of Choice (ZFC), assuming those axioms are consistent. Cardinality, developed primarily from Georg Cantor's work in the late nineteenth century, consequently became one of the foundations of modern set theory and mathematical logic.
Key takeaways
ARTICLE
Cardinality is the mathematical notion used to describe the size of a set, including both finite and infinite sets. For a finite set $A$, its cardinality $|A|$ is simply the number of elements it contains. More generally, two sets $A$ and $B$ have the same cardinality if there exists a bijection between them—a one-to-one correspondence pairing every element of $A$ with exactly one element of $B$. This definition produces a striking feature of infinite sets: a proper subset can have the same cardinality as the whole set. For example, the natural numbers $\mathbb N$ and the even numbers have the same cardinality because $f(n)=2n$ gives a bijection between them.
Infinite sets are divided into countable and uncountable ones. The natural numbers, integers, and rational numbers are countably infinite and have cardinality $\aleph_0$. In contrast, Cantor's diagonal argument shows that the real numbers $\mathbb R$ are uncountable and therefore form a strictly larger infinity. More generally, Cantor's theorem says that for every set $A$, its power set $\mathcal P(A)$ has strictly greater cardinality than $A$, producing an endless hierarchy of increasingly large infinities. Cardinal numbers such as $\aleph_0,\aleph_1,\aleph_2,\ldots$ provide a systematic way of describing these different infinite sizes.
One of the central questions arising from this theory is the Continuum Hypothesis (CH): whether the cardinality of the real numbers is exactly the next cardinal after $\aleph_0$, that is, whether $|\mathbb R|=\aleph_1$. Remarkably, Gödel and Cohen's work ultimately established that CH can neither be proved nor disproved from the standard Zermelo–Fraenkel axioms with the Axiom of Choice (ZFC), assuming those axioms are consistent. Cardinality, developed primarily from Georg Cantor's work in the late nineteenth century, consequently became one of the foundations of modern set theory and mathematical logic.
Key takeaways
- $|A|=|B|$ means that a bijection exists between $A$ and $B$.
- $\mathbb N$, $\mathbb Z$, and $\mathbb Q$ have cardinality $\aleph_0$.
- $\mathbb R$ is uncountable, so $|\mathbb R|>\aleph_0$.
- Cantor's theorem $|A|<|\mathcal P(A)|$ shows that there is no largest infinity.
ARTICLE
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