The sorites paradox goes like this: Start with a heap of sand. Remove a grain of sand and you still have a heap; remove another, and another, and another, and you'll still have a heap. Eventually, though, your heap will be appreciably smaller, and eventually you'll be left with a single grain of sand. So, when exactly did we go from heap to non-heap?
My question is about the real line. (It's not quite the same problem as the sorites paradox, but it's similar enough for the paradox to be illustrative.)
How do we make the move from a dimensionless point to an open interval on the real line (apparently nothing more than a concatenation of dimensionless points) that can be use as measurement? Don't get me wrong: I'm not saying that we start with a point on the real line, then add the "next" one, and ask when it becomes an interval. And I'm not saying that we begin, as with the sorites, with an open interval and successively remove points until we're left (after uncountably many years have passed) with a single point, and ask when the interval became a non-interval.
But in the same way that there's a qualitative (but ambiguous) different between heaps and non-heaps, so there's a difference between points and intervals. How is it that dimensionless points strung together become a one-dimensional interval? And the fact that an open interval of the real line can't be constructed by concatenating points makes the affair seem even more whimsical: instead, intervals are "constructed" by taking two unequal points, considering the span between them (perhaps including the points, as well), and then declaring, by fiat, "Yep--there's an interval there!"
Apologies is this has been asked before. I couldn't find where it had.
Well, that's not quite the same thing, since intervals have a very specific property: If $x$ and $y$ are in the interval and $z$ lies between $x$ and $y$, then $z$ is in the interval as well.
This means that if the interval is open, removing any point would make it no longer an interval. If you removed $1$ from $(0,2)$ you get the union of two intervals, but you don't get an interval.
You might ask, instead, at which point we don't have a set which contains an interval? But then we can just remove all the rational number, or some other countable dense set, then resulting set, $(0,2)\setminus\Bbb Q$ does not contain any interval, since any interval must have a rational number in it. And this process only takes $\aleph_0$ years to complete.
The heap paradox uses the fact that a heap is discrete, and removing a single point does not chain its features very much. Intervals of real numbers don't have this property, and removing a point changes them very much. The question, if you want to transfer it to higher cardinalities, should be asked, perhaps, on a set which looks more like the irrational numbers, or even the rational numbers. But even then, you might want to readjust the notion of "heap" to be something more suitable, rather than just "interval".