The cardinality of the continuum, often denoted by , is the cardinality of the set of real numbers. A set of cardinality is said to have continuum many elements.

Cantor's diagonal argument shows that is uncountable. Furthermore, it can be shown that is equinumerous with the power set of , so . It can also be shown that has uncountable cofinality.

Many properties of are independent of ZFC, that is, they can neither be proved nor disproved in ZFC, assuming that ZF is consistent. For example, for every nonzero natural number , the equality is independent of ZFC. (The case is the well-known Continuum Hypothesis.) The same is true for most other alephs, although in some cases equality can be ruled out on the grounds of cofinality, e.g., . In particular, could be either or , so it could be either a successor cardinal or a limit cardinal, and either a regular cardinal or a singular cardinal.