Editing Beth number (section)

== Relation to the aleph numbers ==
Assuming the [[axiom of choice]], infinite cardinalities are [[total order|linearly ordered]]; no two cardinalities can fail to be comparable. Thus, since by definition no infinite cardinalities are between <math>\aleph_0</math> and <math>\aleph_1</math>, it follows that 
:<math>\beth_1 \ge \aleph_1.</math>
Repeating this argument (see [[transfinite induction]]) yields 
<math>\beth_\alpha \ge \aleph_\alpha</math> 
for all ordinals <math>\alpha</math>.

The [[continuum hypothesis]] is equivalent to
:<math>\beth_1=\aleph_1.</math>

The [[Continuum hypothesis#Generalized continuum hypothesis|generalized continuum hypothesis]] says the sequence of beth numbers thus defined is the same as the sequence of [[aleph number]]s, i.e., 
<math>\beth_\alpha = \aleph_\alpha</math> 
for all ordinals <math>\alpha</math>.