I used "0.(0)" as a non-mathematical somewhat lexical substitution for "infinitely small value"
You were referring to 0.0 repeating, right? That's not an infinitely small values, that's just zero. I went along because initially you seemed to agree that 0.(0)=0, but now you're changing your tune.
Does Dirac delta function exist? Can it be distribution for some number-generator? Is it's integral equal to 1?
Of course, the Dirac delta function is a well-defined probability distribution. There's a perfectly precise and formal mathematical theory that describes exactly what it is and how to integrate it (which of course gives 1). And that same mathematical theory is the one that says there's no uniform distribution over the reals.
What is its pdf in point 1?
The Dirac delta function doesn't have a pdf. Despite its name, it's not actually a function. If we were talking about real-valued functions, rather than probability distributions, and someone tried to bring up the Dirac delta function as a counterexample to some statement about such functions,it would be a complete non-sequitur. Kind of like how bringing up a nonexistent distribution is a non-sequitur.
What i suggested is the process, which is just the reverse of Ddf. You would recieve the same entity by smearing both uniform and normal distributions.
You suggested defining a uniform distribution over the reals as a limit of a sequence of distributions that doesn't have a limit. The difference between this and the Dirac delta function is that the Dirac delta function is the limit of a sequence of distributions that actually do have a limit. You can't just take an arbitrary sequence and expect it to have a limit. I think what you are confused is the difference between a distribution and a function. A distribution is not the same thing as its pdf, and the limit of distributions is not the same thing as the limit of the pdfs. The pdfs of uniform distributions don't have a limit as you shrink the range - they converge to 0 pointwise, except at x=0, where they blow up to infinity. But the distributions themselves do have a limit, the Dirac delta. Meanwhile, the pdfs of uniform distributions have a limit as the range increases (the zero function), but the the distributions themselves do not.
How many points in a [0,1] segment?
How many points in [(0,0); (1,1)] square compared to the previous answer?
Also 2^ℵ₀. It's actually a well-known theorem that the square does not have more points than the line, they have the exact same number. And no, the square isn't a larger infinity - they're both the exact same infinity. Now of course, the square is larger in some ways. It has an area of 1, while the line has an area of 0. But that has nothing to do with the number of points. That's why I asked what you meant by "order of infinity bigger". If you said, "the square is an order of infinity bigger than the line", that doesn't tell me if you're claiming that the square has a larger number of points than the line (which is false), or has a larger area than the line (which is true), or something else, because there is no mathematical concept called "order of infinity bigger". You can't make up terms and then expect me to know what you mean. But I think the reason you did this is because you don't even know what you mean. You have an intuition that one thing is infinitely bigger than another but have no idea how to formalize this intuition, so you threw out a hopelessly vague phrase. If I'm wrong about this, then please clarify by giving an explanation of you meant using accepted mathematical terminology.
I see that you do not have experience with abstract thinking about the concepts I have used (infinitely small values in probability, limit of process, infinite sets comparison), so I will leave this dialog.
If you want to leave the dialog, that is fine, but this is a cowardly way to do it. You're just going to throw out a baseless insult and then try to end the conversation on that note so you don't have to defend it? I think the previous comments speak for themselves that I have a lot more experience than you with abstract reasoning about limits and infinite sets. As for infinitely small values in probability, I have enough experience to know that those aren't actually a thing in standard probability theory, and you don't. And I can guarantee based on your comments that you're not using a formal non-standard probability theory that allows for infinitesimals - you're just running on intuitions that you don't know enough about the actual math to be able to formalize, and you get incorrect conclusions as a result.