The continuum hypothesis says that \aleph_1 = 2^{\aleph_0}, i.e. 2^{\aleph_0} is the smallest cardinal number bigger than \aleph_0, i.e. there is no set whose cardinality is strictly between that of the integers and that of the real numbers. The continuum hypothesis is independent of ZFC, a standard axiomatization of set theory; that is, it is impossible to prove the continuum hypothesis or its negation from ZFC—provided that ZFC is consistent. For more detail, see § Cardinality of the continuum below.
