structural:expansion, stylistic:style_transfer
Successor and limit ordinals

Any nonzero ordinal has the minimum element, zero. It may or may not have a maximum element. For example, 42 has maximum 41 and ω+6 has maximum ω+5. On the other hand, ω does not have a maximum since there is no largest natural number. If an ordinal has a maximum α, then it is the next ordinal after α, and it is called a successor ordinal, namely the successor of α, written α+1. In the von Neumann definition of ordinals, the successor of α is α ∪ { α } {\displaystyle \alpha \cup \{\alpha \}} since its elements are those of α and α itself.

A nonzero ordinal that is not a successor is called a limit ordinal. One justification for this term is that a limit ordinal is the limit in a topological sense of all smaller ordinals (under the order topology).

When ⟨ α ι | ι < γ ⟩ {\displaystyle \langle \alpha _{\iota }|\iota <\gamma \rangle } is an ordinal-indexed sequence, indexed by a limit γ {\displaystyle \gamma } and the sequence is increasing, i.e. α ι < α ρ {\displaystyle \alpha _{\iota }<\alpha _{\rho }} whenever ι < ρ, {\displaystyle \iota <\rho,} its limit is defined as the least upper bound of the set { α ι | ι < γ }, {\displaystyle \{\alpha _{\iota }|\iota <\gamma \},} that is, the smallest ordinal (it always exists) greater than any term of the sequence. In this sense, a limit ordinal is the limit of all smaller ordinals (indexed by itself). Put more directly, it is the supremum of the set of smaller ordinals.

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

So in the following sequence:

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

Thus, every ordinal is either zero, or a successor (of a well-defined predecessor), or a limit. This distinction is important, because many definitions by transfinite recursion rely upon it. Very often, when defining a function F by transfinite recursion on all ordinals, one defines F (0), and F (α+1) assuming F (α) is defined, and then, for limit ordinals δ one defines F (δ) as the limit of the F (β) for all β<δ (either in the sense of ordinal limits, as previously explained, or for some other notion of limit if F does not take ordinal values). Thus, the interesting step in the definition is the successor step, not the limit ordinals. Such functions (especially for F nondecreasing and taking ordinal values) are called continuous. Ordinal addition, multiplication and exponentiation are continuous as functions of their second argument (but can be defined non-recursively).