Editing Limit ordinal (section)

==Examples==
Because the [[Class (set theory)|class]] of ordinal numbers is [[well-order]]ed, there is a smallest infinite limit ordinal; denoted by ω (omega). The ordinal ω is also the smallest infinite ordinal (disregarding ''limit''), as it is the [[least upper bound]] of the [[natural numbers]]. Hence ω represents the [[order type]] of the natural numbers. The next limit ordinal above the first is ω + ω = ω·2, which generalizes to ω·''n'' for any natural number ''n''. Taking the [[union (set theory)|union]] (the [[supremum]] operation on any [[Set (mathematics)|set]] of ordinals) of all the ω·n, we get ω·ω = ω<sup>2</sup>, which generalizes to ω<sup>''n''</sup> for any natural number ''n''. This process can be further iterated as follows to produce:

:<math>\omega^3, \omega^4, \ldots, \omega^\omega, \omega^{\omega^\omega}, \ldots, \varepsilon_0 = \omega^{\omega^{\omega^{~\cdot^{~\cdot^{~\cdot}}}}}, \ldots</math>

In general, all of these recursive definitions via multiplication, exponentiation, repeated exponentiation, etc. yield limit ordinals. All of the ordinals discussed so far are still [[countable]] ordinals. However, there is no [[recursively enumerable]] scheme for [[ordinal notation|systematically naming]] all ordinals less than the [[Church–Kleene ordinal]], which is a countable ordinal.

Beyond the countable, the [[first uncountable ordinal]] is usually denoted ω<sub>1</sub>.  It is also a limit ordinal.

Continuing, one can obtain the following (all of which are now increasing in cardinality):

:<math>\omega_2, \omega_3, \ldots, \omega_\omega, \omega_{\omega + 1}, \ldots, \omega_{\omega_\omega},\ldots</math>

In general, we always get a limit ordinal when taking the union of a nonempty set of ordinals that has no [[maximum]] element.

The ordinals of the form ω²α, for α > 0, are limits of limits, etc.