A simple question.

[Tiberius]

Show me a definition of "infinite". I thought that infinite means something that does not end.

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

An "infinite" number is a number that has infinite value, whilst you are talking about numbers that have a finite value, but are infinitely long. That was the difference. 0.(1) is a finite number, as it has a finite value, but it is infinitely long.

As I've said before, 0.(0)1 is an invalid number; it can't exist. You cannot have an infinite number of 0's and then have a 1 on the end, because there can't be an end for the 1 to go on.

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.

-infinity is a totally different ball park. Both infinity and -infinity are limits of the number line. All numbers have to lie between them. Infinity is positively infinite, whilst -infinity is negatively infinite. I assumed you were talking about positive infinity, since you were talking about lengths of numbers (all of which are positive). I don't see where -infinity comes into it here.

If you want, I'll restate what I said above. 0.11...111 is not positively infinite because I can name a number higher than it (0.2), and it is not negatively infinite because I can name a number lower than it (0).

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.

The problem is you do specify that it ends, because after the ellipsis, you've put 111. That is the end. Notation in this case matters, because if you insist on using the notation 0.11...111, and then saying that the ... means there are an infinite number of 1's there, you are causing a contradiction, because you have an infinite number of 1's followed by "111". This cannot happen.

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).

No, it does not end with a 1. It doesn't end at all. The ... notation implies that the 1's repeat forever, and that the number is infinitely long. As such, it does not have an end.

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

There is no end to this number, it continues for an infinite length, with an infinite number of 0's. In mathematics, it is either equal to 0, or is undefined.

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.

No calculation uses all the digits of pi; we simply use the first few of them in most cases. In this case however, where we want the first actual number to follow 0, your number doesn't fit the bill, since we require every digit.

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.

...and in this case, we want the limit of it, since you gave me a sequence calculation that diverge to 0.

Do you have a better answer?

Yes. The first number that comes after 0 is impossible to calculate, and hence is unknown.