Limit Ordinal

A limit ordinal is an ordinal number which is neither zero nor a successor ordinal.

Another way of defining a limit ordinal is to say that λ is a limit ordinal if and only if:

There is an ordinal less than λ and whenever β is an ordinal less than λ, then there exists an ordinal γ such that β < γ < λ.

So in the following sequence:

0, 1, 2, ..., ω, ω+1

ω is a limit ordinal because for any smaller ordinal (in this example, a natural number) we can find another ordinal (natural number) larger than it, but still less than ω.

Various other ways to define limit ordinal are:

• It is equal to the supremum of all the ordinals below it, but is not zero. (Compare with a successor ordinal: the set of ordinals below it has a maximum, so the supremum is this maximum, the previous ordinal.)
• It is not zero and has no maximum element.
• It can be written in the form ωα for α > 0. That is, in the Cantor normal form there is no finite number as last term, and the ordinal is nonzero.
• It is a limit point of the class of ordinal numbers, with respect to the order topology. (The other ordinals are isolated points.)

Some contention exists on whether or not 0 should be classified as a limit ordinal, as it does not have an immediate predecessor; some textbooks include 0 in the class of limit ordinals while others exclude it.