Because the
class of ordinal numbers is
well-ordered, 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 (the
supremum operation on any
set of ordinals) of all the ω·n, we get ω·ω = ω2, which generalizes to ω
n for any natural number
n. This process can be further iterated as follows to produce: :\omega^3, \omega^4, \ldots, \omega^\omega, \omega^{\omega^\omega}, \ldots, \varepsilon_0 = \omega^{\omega^{\omega^{~\cdot^{~\cdot^{~\cdot}}}}}, \ldots 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
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 ω1. It is also a limit ordinal. Continuing, one can obtain the following (all of which are now increasing in cardinality): :\omega_2, \omega_3, \ldots, \omega_\omega, \omega_{\omega + 1}, \ldots, \omega_{\omega_\omega},\ldots 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. == Properties ==