On Feb 1, 2014, at 2:05 PM, Joe Niederberger <firstname.lastname@example.org> wrote:
> You don't need to know everything about real numbers to follow my interpretation, and arrive at an understanding that (1) lines today are (usually) viewed as sets of points, (2) the cut point has to belong to one subset or the other but not both, assuming (3) you want exactly two subsets that are disjoint and when unioned give back the original. > > That much is very elementary, naive, set theory. Sure there's that can be said, but one step at a time, huh? > You don't need to get into density and limits and whatnot > to look at what's going with the cut point itself. > > I arrived at this by reading Neighbor's questions and trying honestly to see what he was after. I could be wrong.
I guess I felt that Neighbor was asking - What do you mean by ?set of points?? What are these points? If they have no width then how can they add up to make a line, even an infinite number of them? And so forth. I was just thinking that he might be able to apply what you wrote regarding ?sets" to "points on a line" if he understood "points on a line" better. The best model I am aware of to do that with is the set of real numbers. I suppose that you could make headway with just the rational numbers, as long as the student has given them sufficient thought, as a set. I am not saying that he should know what density means formally but he can get a feel for it. I don?t even know what density means formally, but I have a good feel for it and if I read the definition I would probably say to myself ?Ok, that makes sense."