Relation To The Aleph Numbers
Assuming the axiom of choice, infinite cardinalities are linearly ordered; no two cardinalities can fail to be comparable. Thus, since by definition no infinite cardinalities are between and, it follows that
Repeating this argument (see transfinite induction) yields for all ordinals .
The continuum hypothesis is equivalent to
The generalized continuum hypothesis says the sequence of beth numbers thus defined is the same as the sequence of aleph numbers, i.e., for all ordinals .
Read more about this topic: Beth Number
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