Editing Limit ordinal (section)

== Properties ==
The classes of successor ordinals and limit ordinals (of various [[cofinality|cofinalities]]) as well as zero exhaust the entire class of ordinals, so these cases are often used in proofs by [[transfinite induction]] or definitions by [[transfinite recursion]]. Limit ordinals represent a sort of "turning point" in such procedures, in which one must use limiting operations such as taking the union over all preceding ordinals.  In principle, one could do anything at limit ordinals, but taking the union is [[continuous function (topology)|continuous]] in the order topology and this is usually desirable.

If we use the [[von Neumann cardinal assignment]], every infinite [[cardinal number]] is also a limit ordinal (and this is a fitting observation, as ''cardinal'' derives from the Latin ''cardo'' meaning ''hinge'' or ''turning point''): the proof of this fact is done by simply showing that every infinite successor ordinal is [[equinumerous]] to a limit ordinal via the [[Hilbert's paradox of the Grand Hotel|Hotel Infinity]] argument.

Cardinal numbers have their own notion of successorship and limit (everything getting upgraded to a higher level).