Editing Cardinal number (section)

== The continuum hypothesis ==
The [[continuum hypothesis]] (CH) states that there are no cardinals strictly between <math>\aleph_0</math> and <math>2^{\aleph_0}.</math> The latter cardinal number is also often denoted by <math>\mathfrak{c}</math>; it is the [[cardinality of the continuum]] (the set of [[real number]]s). In this case <math>2^{\aleph_0} = \aleph_1.</math>

Similarly, the [[generalized continuum hypothesis]] (GCH) states that for every infinite cardinal <math>\kappa</math>, there are no cardinals strictly between <math>\kappa</math> and <math>2^\kappa</math>. Both the continuum hypothesis and the generalized continuum hypothesis have been proved to be [[Independence (mathematical logic)|independent]] of the usual axioms of set theory, the Zermelo–Fraenkel axioms together with the axiom of choice ([[Zermelo–Fraenkel set theory|ZFC]]).

Indeed, [[Easton's theorem]] shows that, for [[regular cardinal]]s <math>\kappa</math>, the only restrictions ZFC places on the cardinality of <math>2^\kappa</math> are that <math> \kappa < \operatorname{cf}(2^\kappa) </math>, and that the exponential function is non-decreasing.