cardinality of the continuum

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

Cantor's diagonal argument shows that $\continuum$ is uncountable. Furthermore, it can be shown that $\R$ is equinumerous with the power set of $\N$ , so $\continuum=2^{\aleph_0}$ . It can also be shown that $\continuum$ has uncountable cofinality.

It can also be shown that $$\continuum=\continuum^{\aleph_0}=\aleph_0\continuum=\continuum\continuum =\continuum+\kappa=\continuum^n$$ for all finite cardinals $n\ge1$ and all cardinals $\kappa\le\continuum$ . See the article on cardinal arithmetic for some of the basic facts underlying these equalities.

There are many properties of $\continuum$ that 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 $n$ , the equality $\continuum=\aleph_n$ is independent of ZFC. (The case $n=1$ 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., $\continuum\neq\aleph_\omega$ . In particular, $\continuum$ could be either $\aleph_1$ or $\aleph_{\omega_1}$ , so it could be either a successor cardinal or a limit cardinal, and either a regular cardinal or a singular cardinal.