I never guaranteed you that the process will finish. In fact I agree it will not, since it is infinite.
However, each subprocess within the process is finite and has a definite end. That is why the process is countable, albeit indefinite.
> No difference if you are trying to count the naturals, the reals or the digits of pi.
There is a difference. You can count the naturals. Each subprocess is finite: 1, 5, 2, 6, 3, 7, ....
You can count the digits of pi: 3, 1, 4, 1, 5, 9, ....
You cannot count the reals. randomreal() is not guaranteed to be finite a finite subprocess. I cannot guarantee you that any given invocation of randomreal() will terminate. In fact, it is trivial to construct a subprocess sequence where it fails to terminate:
randomreal(1) = 1
randomreal(2) = 3.14159.....
We will never reach randomreal(3).
In that sense, your issue is that your view is self-contradictory. For it to be consistent, you need to stop talking about the reals, because there is no such thing as the reals. Or, accept the notion of there being an infinite number of alephs.
You cannot both reference the reals and simultaneously claim there are not an infinite number of alephs without contradicting yourself. This is not a minor detail.
Which you path you choose, as far as I'm aware, is currently unknowable and a matter of personal philosophy rather than objectivity.
How I can you “guarantee” that it will finish?
You can’t. You are assuming a determinate result from an infinite process.
No difference if you are trying to count the naturals, the reals or the digits of pi.