What is a sequence of ordinals?
The concept of a sequence of ordinals shows up here and here, and in the definition of cofinality in Jech, third edition, page 31.
$\text{cf}(\alpha) =$ the least limit ordinal $\beta$ such that there is an increasing $\beta$ sequence $\langle \alpha_\xi : \xi < \beta \rangle $ with $\lim_{\xi \to \beta} \alpha_\xi = \alpha $.
For what it's worth, I found the Kunen definition of cofinality here to be easier to understand, reproduced below. $\text{type}$ is the order type. (Basically, we're looking at the powerset of $\gamma$ and seeing how big of a powerset we need to limit to $\gamma$ itself. I think the Jech definition has the same idea, we're just ordering the subsets of $\gamma$ with their intrinsic ordering).
$$ \text{cf}(\gamma) = \min \{ \text{type}(x) : x \subset \gamma \land \sup(x) = \gamma \} $$
Anyway, I think part of the reason that I was initially thrown off by the Jech definition is that I think of sequences as being functions $\mathbb{N} \to X$ for some set $X$.
A sequence is this setting is clearly more general, possibly a function $\kappa \to X$ for some ordinal $\kappa$.
There are other choices we could make though: we could restrict $\kappa$ to be a cardinal, we could restrict $\kappa$ to be a limit ordinal, or we could allow $\kappa$ to be any order type, ordinal or not. I could also insist that $\kappa$ is a directed set and make myself a nice net.
What do people commonly use a sequence of ordinals to mean? What's the most restrictive definition I could take for starting out with set theory that's still broad enough to include sequences I'm likely to encounter?