Successor and limit ordinals

Let $\alpha$ be an ordinal (Ordinal (von Neumann)).

• $\alpha$ is a successor ordinal when $\alpha = \beta^{+} = \beta \cup \{\beta\}$ for some ordinal $\beta$, which is then an ordinal by Basic closure properties of ordinals;
• $\alpha$ is a limit ordinal when $\alpha \ne 0$ and $\alpha$ is not a successor ordinal.

Every ordinal is therefore exactly one of: $0$, a successor ordinal, or a limit ordinal. The three cases are the three clauses of a definition or a proof by transfinite recursion or induction over the ordinals.

• The predecessor of a successor is unique. If $\alpha = \beta^{+}$ then $\beta \in \alpha$ and every $\xi \in \alpha$ satisfies $\xi \le \beta$, so $\beta$ is the largest element of $\alpha$ and is determined by $\alpha$. In particular $\beta^{+} = \gamma^{+}$ forces $\beta = \gamma$.
• Union characterisation. For a nonzero ordinal $\alpha$: $\alpha$ is a limit ordinal if and only if $\alpha = \bigcup \alpha$, and $\alpha$ is a successor if and only if $\bigcup \alpha \in \alpha$, in which case $\alpha = (\bigcup \alpha)^{+}$. For the successor case, $\bigcup(\beta \cup \{\beta\}) = (\bigcup \beta) \cup \beta = \beta$, because $\bigcup \beta \subseteq \beta$ by transitivity. For the limit case, $\bigcup \alpha \subseteq \alpha$ always holds by transitivity, and conversely, given $\xi \in \alpha$, the ordinal $\xi^{+}$ satisfies $\xi^{+} \in \alpha$: by Trichotomy and well-ordering of the ordinals the alternatives are $\xi^{+} = \alpha$, excluded because $\alpha$ is not a successor, and $\alpha \in \xi^{+}$, which gives $\alpha \in \xi$ or $\alpha = \xi$ and hence $\alpha \in \alpha$ using $\xi \in \alpha$ and transitivity, excluded by Basic closure properties of ordinals; so $\xi \in \xi^{+} \in \alpha$ puts $\xi \in \bigcup \alpha$. The hypothesis $\alpha \ne 0$ cannot be dropped, since $\bigcup 0 = 0$.
• Closure under successor. The previous paragraph says exactly that a nonzero ordinal is a limit if and only if it is closed under the successor operation. That is the form in which limit ordinals are recognised in practice.
• $0$ is not a limit ordinal here. Some texts include it, so that "limit ordinal" means "$\alpha = \bigcup \alpha$" outright. The convention adopted is the more widely used one, and it is the one that makes "$0$, successor, limit" a genuine three way split.
• The least limit ordinal is $\omega$ ($\omega$ is the least limit ordinal), so the distinction is invisible below $\omega$: every natural number is either $0$ or a successor. That is precisely why ordinary induction on $\mathbb{N}$ needs only a base case and a successor step, while induction over the ordinals needs a limit clause as well.