Cardinality of the continuum

Definition

The cardinality of the continuum is a term used for an infinite cardinal defined in the following equivalent ways:

1. It is the cardinality of the set of real numbers.
2. It is the power cardinal corresponding to the smallest infinite cardinal. Thus, it is the Beth number Beth one.
3. It is the cardinality of the closed unit interval $[0,1]$.
4. It is the cardinality of any finite-dimensional Euclidean space.
5. It is the cardinality of any (finite-dimensional) manifold.

Assuming the continuum hypothesis

The continuum hypothesis states that the cardinality of the continuum is the smallest uncountable cardinal, i.e., it equals $\aleph _{1}$. The continuum hypothesis is independent of ZFC (the standard axiomatic framework of set theory) but it follows from the axiom of constructibility. It is not generally considered to be either true or false.