Definition 1. The successor of an ordinal $\alpha$ is the ordinal
\[\alpha + 1 = \alpha \cup \{\alpha\}.\]An ordinal $\alpha$ is a successor ordinal if $\alpha = \beta+1$ for some ordinal $\beta$.
Proposition 1. For any ordinal $\alpha$,
\[\alpha + 1 = \inf \{\beta : \beta > \alpha\}.\]Proposition 2. For any infinite ordinal $\alpha$,
\[\lvert \alpha + 1 \rvert = \lvert \alpha \rvert.\]Proof.
\[f : \alpha + 1 \to \alpha, \, \xi \mapsto \begin{cases} 0, & \text{if $\xi = \alpha$}; \\ \xi + 1, & \text{if $\xi < \omega$}; \\ \xi, & \text{otherwise} \end{cases}\]is an injection.