Sequence

Definition 1. A transfinite sequence is a function whose domain is an ordinal. If $s : \alpha \to X$ is transfinite sequence for some ordinal $\alpha$:

• $s$ is an $\alpha$-sequence.

• $\alpha$ is the length of $s$.

• $s$ is a transfinite sequence in $X$.

• $s$ is also denoted by

  \[\langle s_\xi : \xi < \alpha \rangle \quad\text{or}\quad \langle s_\xi \rangle_{\xi < \alpha},\]

  where $s_\xi$ stands for $s(\xi)$.

Definition 2. A countably infinite sequence is an $\omega$-sequence. A countably infinite sequence $s$ is also denoted by

\[\langle s_n : n \in \N \rangle \quad\text{or}\quad \langle s_n \rangle_{n \in \N}.\]

Definition 3. A finite sequence is an $n$-sequence for some natural number $n$.

Definition 4. If $s$ is an $\alpha$-sequence for some ordinal $\alpha$, the extension of $s$ by $x$ is

\[s^\frown x = s \cup \{(\alpha,x)\}.\]