I just don’t see what describability, or as jiggawatts calls it, naming, has to do with AC. AC is built upon an indexed set[1].
That way you can just reference the index. It seems extremely contrived to me to argue that being finitely nameable has any bearing on underlying mathematical principle, nothing about any form of infinity is relatable to the physical confines of this universe, of which language is a member, but the concept of infinite is very useful.