Two sets $A$ and $B$ are equinumerous (written $A \sim B$ or $|A| = |B|$) if there exists a bijection $f: A \to B$. This is an equivalence relation on sets. The cardinality $|A|$ of a set $A$ is (assuming AC) the least ordinal equinumerous with $A$.

An ordinal $\kappa$ is a cardinal (or initial ordinal) if there is no bijection between $\kappa$ and any smaller ordinal. Equivalently, $\kappa$ is a cardinal if $|\kappa| = \kappa$. The finite cardinals are $0, 1, 2, \ldots$, and the infinite cardinals are:

$\aleph_0 = \omega, \quad \aleph_1 = \omega_1, \quad \aleph_2 = \omega_2, \quad \ldots, \quad \aleph_\omega, \quad \ldots$

where $\omega_\alpha$ is the $\alpha$-th infinite initial ordinal.

For cardinals $\kappa, \lambda$:

• Cardinal sum: $\kappa + \lambda = |A \sqcup B|$ where $|A| = \kappa$, $|B| = \lambda$, $A \cap B = \emptyset$.
• Cardinal product: $\kappa \cdot \lambda = |A \times B|$.
• Cardinal exponentiation: $\kappa^\lambda = |A^B|$ (the set of all functions $B \to A$).

For infinite cardinals: $\kappa + \lambda = \kappa \cdot \lambda = \max(\kappa, \lambda)$ (assuming AC).