Limit Ordinal - Examples

Examples

Because the class of ordinal numbers is well-ordered, there is a smallest infinite limit ordinal; denoted by ω. This 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, and then we have 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. This process can be iterated as follows to produce:

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):

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

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