cardinality

Cardinality is the precise notion of the “size” of a set, defined not by counting but by comparison through functions. Two sets have the same cardinality when there is a bijection between them; one has cardinality at most that of another when there is an injection into it. This definition needs no numbers at all, so it extends gracefully from finite collections to infinite ones, where ordinary counting breaks down.

For finite sets cardinality is just the number of elements, |A| = n. For infinite sets it becomes a genuinely new idea with surprises: the naturals, integers, and rationals all share the same cardinality, written ℵ0 (aleph-naught), the smallest infinite cardinal; while the reals have the strictly larger cardinality of the continuum, c = 2^(ℵ0). The Cantor–Schröder–Bernstein theorem reassures us that the ordering behaves sensibly: if each of two sets injects into the other, they have equal cardinality.

Cantor's theorem shows the cardinals never stop climbing: for every set A, its power set has strictly greater cardinality than A, so there is no largest infinity. A famous open-ended subtlety is the continuum hypothesis — whether any cardinal lies strictly between ℵ0 and c — which Gödel and Cohen showed can neither be proved nor disproved from the standard axioms. In analysis, cardinality explains why the rationals are negligible (measure zero) while the irrationals fill the line.