>>12238343Good question, don't listen to the other moron. Points measure zero dimensional things (like discrete sets of points, in which case "measure" means "count.") Lines measure 1-dimensional things.
How is it possible that a line can do this? It's actually surprising that it is. This is a field of math called measure theory, which people usually don't learn until late into undergrad. Basically the way it works is you say, well I want the interval [a, b] to have length b-a. So you define that to be the length of that interval. So what's the length of any other set of real numbers? You take the set, cover it in a sequence of intervals like [a,b], and take finer and finer covers (more precise). each time you compute the total length of the cover. the limit of this sequence of lengths is the length of your set.
turns out this is really hard to justify as a good notion of length (e.g, if a set contains a subset, the subset is shorter, or, if you split a set into two pieces, the lengths add up, lots of little things like that) it's called "lebesgue measure."
so lebesgue measure has nothing to do with points, or how close those points are to one another. in fact, without starting with the length of intervals we'd have no idea what to do. this is because all the points that make up the line each have lebesgue measure 0. this is what you probably mean when you say they're zero dimensional. then anything with countably many points has lebesgue measure zero.
but a line has uncountably many points, which is a lot bigger. the whole point of measure theory is finding a way around the issue that a bunch of length zero things come together to form a length 1 thing, and this issue will never be solved if you try to start with points instead of starting with intervals.
but solving this issue comes with other problems. now because of all this uncountability business there's room to create sets which are so nasty that there is no way to assign a length to them.
