A simple question.

[Zenith]

1) 0.11...111 would be finite even if it didn't end with a 1, since I can name a number higher than it (0.2), and hence it is not infinite. What you mean is that it is finitely long, which is perfectly true if it ends with a 1, and the way you have written it (0.11...111) shows this.

Show me a definition of "infinite". I thought that infinite means something that does not end.
This is what I've found about finite:

1. having bounds or limits; not infinite; measurable.
2. Mathematics .
a. (of a set of elements) capable of being completely counted.
b. not infinite or infinitesimal.
c. not zero.
3. subject to limitations or conditions, as of space, time, circumstances, or the laws of nature: man's finite existence on earth.

And infinite:

3. unlimited or unmeasurable in extent of space, duration of time, etc.: the infinite nature of outer space.
4. unbounded or unlimited; boundless; endless: God's infinite mercy.
5. Mathematics .
a. not finite.
b. (of a set) having elements that can be put into one-to-one correspondence with a subset that is not the given set.

So by this notion 0.(1) is not a finite number and 0.(0)1 is not a finite number.

Also, you said:
" 0.11...111 would be finite even if it didn't end with a 1, since I can name a number higher than it (0.2), and hence it is not infinite"
So I guess that -infinity is a finite number because even -10^10000000000000 is greater than it.

2) 0.(1) is not the same as 0.11...111, since by definition 0.(1) doesn't have an end, and 0.11...111 has a very visible end. The correct notation for 0.(1) is 0.111... (the ellipsis indicating that the 1 repeats endlessly).

I guess it's just a matter of notation. If I say that 0.11...111 has 234 decimals then yes, it is finite. But if I specify that it never ends, you can't put a real end to it. I know the notation "0.111..." is used to show an infinite number of decimals. But that's just a notation.

3) Whilst 0.11...111 does end with a very visible 1, 0.(1) or 0.111... does not. The ellipsis makes it endless, and so whilst it is a finite value, it is not of finite length.

Now it's a funny thing: if 0.111... does not end with a very visible 1, it doesn't mean that it can be other digit. So it's clearly "1", only that it's not shown (visible).

Your example does have an end: 0.00...001 <-- there it is.

Show me an end to this number: 1/10 * 1/100 * 1/1000 ...

Whilst your calculation is a valid one, the answer to it diverges to 0 as I pointed out, and as such it is not a valid answer to the question "what is the first number that comes after 0?" because either (a) it eventually becomes 0, or (b) it diverges to a value which is impossible to calculate (since we would need a infinite amount of time in order to do it).

Well, as far as I know, not even PI is possible to be calculated (with all decimals), but that doesn't make it an invalid number.

I never insisted that your calculation must not be used, I said that the calculation diverges to 0, which in mathematics either means it becomes equal to 0, or becomes infinitely small and cannot be calculated.

Not even PI can be calculated (to have the exact number). As about my number, it can practically become 0 only if you calculate the lim() of it.

I asked you what the first number to come after 0 was, and you have not given me a valid answer.

Do you have a better answer?