Editing Continuum hypothesis (section)

===Implications of GCH for cardinal exponentiation===
Although the generalized continuum hypothesis refers directly only to cardinal exponentiation with 2 as the base, one can deduce from it the values of cardinal exponentiation <math>\aleph_{\alpha}^{\aleph_{\beta}}</math> in all cases. GCH implies that for ordinals {{mvar|α}} and {{mvar|β}}:{{r|HaydenKennison1968}}

*<math>\aleph_{\alpha}^{\aleph_{\beta}} = \aleph_{\beta+1}</math> when {{math|''α'' ≤ ''β''+1}};
*<math>\aleph_{\alpha}^{\aleph_{\beta}} = \aleph_{\alpha}</math> when {{math|''β''+1 < ''α''}} and <math>\aleph_{\beta} < \operatorname{cf} (\aleph_{\alpha})</math>, where '''cf''' is the [[cofinality]] operation; and
*<math>\aleph_{\alpha}^{\aleph_{\beta}} = \aleph_{\alpha+1}</math> when {{math|''β''+1 < ''α''}} and {{nowrap|<math>\aleph_{\beta} \ge \operatorname{cf} (\aleph_{\alpha})</math>.}}

The first equality (when {{mvar|''α'' ≤ ''β''+1}}) follows from:

<math display="block">\aleph_{\alpha}^{\aleph_{\beta}} \le \aleph_{\beta+1}^{\aleph_{\beta}} =(2^{\aleph_{\beta}})^{\aleph_{\beta}} = 2^{\aleph_{\beta}\cdot\aleph_{\beta}} = 2^{\aleph_{\beta}} = \aleph_{\beta+1} </math>
while:

<math display="block">\aleph_{\beta+1} = 2^{\aleph_{\beta}} \le \aleph_{\alpha}^{\aleph_{\beta}} .</math>

The third equality (when {{mvar|''β''+1 < ''α''}} and <math>\aleph_{\beta} \ge \operatorname{cf}(\aleph_{\alpha})</math>) follows from:

<math display="block">\aleph_{\alpha}^{\aleph_{\beta}}  \ge \aleph_{\alpha}^{\operatorname{cf}(\aleph_{\alpha})} > \aleph_{\alpha} </math>

by [[Kőnig's theorem (set theory)#Kőnig's_theorem_and_cofinality|Kőnig's theorem]], while:

<math display="block">\aleph_{\alpha}^{\aleph_{\beta}} \le \aleph_{\alpha}^{\aleph_{\alpha}} \le (2^{\aleph_{\alpha}})^{\aleph_{\alpha}} = 2^{\aleph_{\alpha}\cdot\aleph_{\alpha}} = 2^{\aleph_{\alpha}} = \aleph_{\alpha+1}</math>