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Actually, to be pedantic, the probability is one. Not "very close to one", but "exactly one".

P (X | X) = 1.

Yep. That's why one speaks of probability density functions and not just probability functions.

I want your random number generator.

Good thing telescopes aren't 0 in size.

It is not equal to zero. It is infinitesimal. There is a difference.

Infinitesimals don't exist in standard analysis. There exist no positive numbers smaller than all positive real numbers. The probability of picking any given real number from [0, 1) is exactly zero, there's no other number it could be. Rigorously, the measure of any countable subset of real numbers is zero.

Honestly, I don't really "like" real numbers, exactly because they are defined as classes of equivalence of converging sequences of neighborhoods.

But I'm pretty sure even in rational numbers probability would be exactly zero, because that's how probability there is defined (as a limit of a sequence, and limit is zero).