Cardinality
Cardinality is the measure of the number of elements in a set, extending the concept of "size" to infinite sets.
Formula
|A| = \text{number of elements in } A
Definition
Cardinality is just the count of how many elements are in a set, the "size" of a set; for finite sets this matches ordinary counting, but the concept becomes fascinating and strange for infinite sets. Two sets have the same cardinality if there is a bijection (one-to-one correspondence) between them; $|\mathbb{N}| = |\mathbb{Z}| = |\mathbb{Q}|$ (all countably infinite, denoted $\aleph_0$), but $|\mathbb{R}| > |\mathbb{N}|$ (uncountable). Formally, cardinality is an equivalence class of sets under bijection; the Cantor-Bernstein-Schroeder theorem says if $|A| \le |B|$ and $|B| \le |A|$ then $|A| = |B|$, and the cardinal hierarchy continues $\aleph_0, \aleph_1, \ldots$, with the Continuum Hypothesis ($|\mathbb{R}| = \aleph_1$) independent of ZFC.
Example
$|\{a, b, c\}| = 3$, $|\{1, 2, \ldots, 100\}| = 100$, $|\emptyset| = 0$, just counting the elements. The bijection $n \mapsto 2n$ matches each natural number to an even natural number, so $|\mathbb{N}| = |\text{even naturals}|$, a surprising result for infinite sets, and Cantor's diagonal argument proves $|\mathbb{R}| > |\mathbb{N}|$ because no list can exhaust all real numbers. Even more surprisingly, $|\mathbb{R}| = |\mathbb{R}^2| = |\mathbb{R}^n|$ for all finite $n$, meaning the plane and the line have the same cardinality, a result that shocked Cantor himself.
Key Insight
For finite sets, cardinality is simply counting, but infinite cardinality reveals that "more elements" is not the whole story. The independence of the Continuum Hypothesis (proved by Godel and Cohen) shows there are genuinely irresolvable questions in mathematics, even about something as basic as the size of the real numbers.