Infinite Sequence vs. Indexed Set

I'm having a look at this provable cryptography tutorial and early on there is a definition of something called a "probability ensemble" which I haven't come across before.

A probability ensemble $X = \{X(a,n)\}_{a \in \{0, 1\}^*;n\in \mathbb N}$ is an infinite sequence of random variables indexed by $a \in \{0, 1\}^*$ and $n \in \mathbb N$.

I just want to make sure I fully understand this. Is "sequence" here being used is a more general sense to mean something like "indexed set"? Otherwise I'm slightly confused as to how this can be a sequence. My understanding of a sequence is that it's formally a bijection from the natural numbers - and the above looks like it's uncountable due to the binary word index.

Can anyone clarify this? I can't easily find other definitions of probability ensemble to compare with so if anyone has any insight I'd be very grateful.

Granted, "sequence" is a little bit of a misnomer but the indexing set is countable: we have a random variable for every pair of $\alpha \in \{0,1\}^\ast$ (i.e. a finite string from the alphabet $\{0,1\}$) and $n \in \Bbb N$. So there is also some (implicit?) probability space on which the functions $X(\alpha,n)$ are defined (and I suppose the values are in $\Bbb R$).

Get used to some casualness in notations in cryptography literature.. Many things are just assumed to well-known or obvious, sometimes. It's good to stay awake. Keep asking, no harm in that.