Continuous distribution and probability in specific point

Question

The key difference between between discrete and continuous variable (or distribution) is that continuous variables cannot be counted. So, if I have time interval from 0 to 1 seconds, in between there are milliseconds, microseconds, nanoseconds etc. It never ends, so I cannot count all the values. Further, we say that probability in one specific point of continuous distribution is 0, for example, P(X=0.25) = 0. What confuses me is that if I could not count all possible values in the continuous distribution, how can I define specific point (e.g., X=0.25)? In other words, if I can say something about specific point, then I can count all specific points. But we just said that I cannot. How to resolve this paradox?

Thanks

Answer 1

Note that the set of real numbers is uncountable and so, your claim that "if I can say something about specific point, then I can count all specific points" is not correct. Indeed, you can claim that $P(x)=0$ for any $x$ in your continuous range of points, but you cannot count all such points.

Answer 2

What does it mean that you can't count them? You can select one specific point. Then you can select another. And another. And another. So, if I understand correctly, you say you can keep doing that until you get through them all.

But here's the catch: even if you go on for infinity, you'll never go through them all. You'll have specified infinity different numbers. But that'll be an incredibly small part of all real numbers: even if you repeat that procedure infinitely many times, you'll not make any real progress.

That said, you can specify numbers by properties. For example, "the positive one that, if multiplied by itself, gives 7" is $\sqrt{7}$. You can infinitely subdivide ranges of numbers, but the number $\sqrt{7}$ is just one.