Editing Limit cardinal (section)

== Constructions ==

One way to construct limit cardinals is via the union operation: <math>\aleph_{\omega}</math> is a weak limit cardinal, defined as the union of all the alephs before it; and in general <math>\aleph_{\lambda}</math> for any [[limit ordinal]] ''λ'' is a weak limit cardinal.

The [[beth number|ב operation]] can be used to obtain strong limit cardinals.  This operation is a map from ordinals to cardinals defined as
:<math>\beth_{0} = \aleph_0,</math>
:<math>\beth_{\alpha+1} = 2^{\beth_{\alpha}},</math> (the smallest ordinal [[equinumerous]] with the powerset)
:If ''&lambda;'' is a limit ordinal, <math>\beth_{\lambda} = \bigcup \{ \beth_{\alpha} : \alpha < \lambda\}.</math>
The cardinal
:<math>\beth_{\omega} = \bigcup \{ \beth_{0}, \beth_{1}, \beth_{2}, \ldots \} = \bigcup_{n < \omega}  \beth_{n} </math>
is a strong limit cardinal of [[cofinality]] &omega;. More generally, given any ordinal ''&alpha;'', the cardinal
:<math>\beth_{\alpha+\omega} = \bigcup_{n < \omega} \beth_{\alpha+n} </math>
is a strong limit cardinal. Thus there are arbitrarily large strong limit cardinals.