Limit ordinal


In set theory, a limit ordinal is an ordinal number that is neither zero nor a successor ordinal. Alternatively, an ordinal λ is a limit ordinal 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. Every ordinal number is either zero, or a successor ordinal, or a limit ordinal.
For example, ω, the smallest ordinal greater than every natural number is a limit ordinal because for any smaller ordinal n we can find another natural number larger than it, but still less than ω.
Using the Von Neumann definition of ordinals, every ordinal is the well-ordered set of all smaller ordinals. The union of a nonempty set of ordinals that has no greatest element is then always a limit ordinal. Using Von Neumann cardinal assignment, every infinite cardinal number is also a limit ordinal.

Alternative definitions

Various other ways to define limit ordinals are:
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.

Examples

Because the class of ordinal numbers is well-ordered, there is a smallest infinite limit ordinal; denoted by ω. The ordinal ω is also the smallest infinite ordinal, 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 of all the ω·n, we get ω·ω = ω2, which generalizes to ωn for any natural number n. This process can be further 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 :
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

The classes of successor ordinals and limit ordinals 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 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 : the proof of this fact is done by simply showing that every infinite successor ordinal is equinumerous to a limit ordinal via the Hotel Infinity argument.
Cardinal numbers have their own notion of successorship and limit.