How to define infinite lists?

Question

Suppose we are working over some universe set $A$. I am interested in how to precisely define a set of infinite lists/sequences only, whose elements are members of $A$.

In the case of finite lists, we could perhaps use notation $a_i^n$ to stand for a (finite) list of length $n \in \mathbb{N}$ where $1 \leq i \leq n$ and each $a_i \in A$.

But say we are only interested in infinite lists. Having computer science background, I often find in literature notation $a_i^{\omega}$ denoting infinite lists of members of $A$. There is no explicit explanation for what $\omega$ stands for.

Could it be that $\omega$ stands for the first infinite ordinal, is that the solution to my question? If so, what about other larger ordinals, should they also be included in the definition? Does this question even makes sense, i.e., should I define what I exactly mean by infinite?

(As my understanding of cardinals vs. ordinals together with this post suggests, cardinals cannot be used as a solution to my problem)

Answer 1

By list we naively think about sequences indexed by natural numbers, or their transfinite extension by ordinals.

So a list is just a function from some ordinal into some set.

To your question, $\omega$ is the first infinite ordinal, which corresponds to the natural numbers. Other ordinals represent longer sequences.

So yes, an infinite list of length $\omega$ is a function from $\omega$ to $A$. If you want to consider also longer lists, these would correspond to functions from larger ordinals into $A$.

Answer 2

$\omega$ is standard notation for the first infinite ordinal, denoting the order type of the natural numbers under their usual ordering. With the usual Von Neumann ordinals, this is represented by the set $\{0,1,2,3,\ldots\}$ -- that is, $\omega$ is just a fancy name for $\mathbb N$.

The notation $A^\omega$ stands for the set of all functions from $\omega$ into $A$ -- and you should be able to convince yourself that a function from $\omega$ to $A$ is exactly what an infinite list of elements of $A$ is. (In the computer science tradition we can also write $\omega\to A$ instead of $A^\omega$, but this usage is not common among mathematicians).

For a single particular list, I don't think writing $a_i^\omega$ would be understood. I would write something like $(a_n)_{n\in\mathbb N}$ instead.