R Hansen says: >When young students split a segment into two segments, they don?t meet condition (d). They create two complete (closed) segments which meet conditions a, b, and c, but not d.
Well, when the Greeks "bisected" (rather than split) they did the same, the bisected segments had a point in common. But remember Neighbor asked for two lines that "could be taken separately, without any problem". When I pressed him further on what that meant, he agreed that he wanted "disjoint" lines -- something the Greeks didn't do or worry about (as far as I know). I don;t know if Neighbor and I even agree on what "disjoint" means, but the discussion of set theory is to at least give him my meaning, he can agree to it or tell me he means something else again.
R Hansen says: >(1) and (2) are not ?segments? that I am aware of (and you say that later).
You are right. Intervals then. Segments in Euclidean Geometry would always have endpoints I think.
Going all the way back to Neighbor's original questioning, he seemed to want to understand what's going on at a fine grained point-by-point level. If I'm wrong about that, this has all been a big misunderstanding on my part.
R Hansen says: >And how does this teach children about points, lines and infinity?
Well, we could go on from here. The important thing is to adopt the language and operations of set theory, if you want to understand how today's mathematicians understand basic facts about points, lines and such, which is what I thought Neighbor was after.
And why for heaven's sake, do *you* keep thinking I'm talking about making lesson plans for a 12 year old? That's your baggage that you are bringing to this discussion. That being said though, a 12 year can surely understand sets, subsets, unions, intersections, etc. Naive set theory. I wouldn't give them Halmos' book, but they could surely have been given a start at 12. The above however, was written for Neighbor, in the context of his questions. Even if I'm wrong about his intentions, if he *did* work through it, then I think his further questions might be clearer to us, hence the fog starts to dissipate, even if just a little.