I don't think this is a slam dunk. For this argument to work, the dart probability must be 100% for any function. This is supposed to be clear "intuitively", and then, by constructing a counterexample using the CH, it's concluded that the CH is false.
But the space of functions from R to countable subsets of R is so vast (and so far removed from the physical world) that I don't think it's possible to have any "intuition" of what's possible in that space. And indeed, we see that there's a construction of a function f that doesn't conform to the "intuition". If there's an "intuitive" line of reasoning and a formal one, and they disagree, shouldn't we just conclude that our intuition is flawed?
> shouldn't we just conclude that our intuition is flawed?
Alternatively, we might conclude that our intuition is right and instead our definition of real numbers isn't exactly what we want for some cases/questions.
But the space of functions from R to countable subsets of R is so vast (and so far removed from the physical world) that I don't think it's possible to have any "intuition" of what's possible in that space. And indeed, we see that there's a construction of a function f that doesn't conform to the "intuition". If there's an "intuitive" line of reasoning and a formal one, and they disagree, shouldn't we just conclude that our intuition is flawed?