> "Two consecutive points" has no meaning in the real > number based > continuum, or even in the set of rational numbers > with usual ordering. So, if you cannot accept that > between two real (or rational) numbers there is > always another (which means that "two consecutive" > simply does not exist in these games) then you will > never accept the more or less accepted answers to > your questions.
You are mentioning a straight-line with numbers on it, and since you can always find a number between two others, being this number on the straight-line, between two numbers on the straight-line there will be another, but I want you to explain me the nature of a straight-line in Euclid's geometry, not putting numbers on the straight-line, and not saying that what for numbers is true is also true for straight-lines. Just now I want to deal only with straight-lines and points on them, as Euclid does in the Elements, without putting numbers on them.
> So, lets simplify a bit: can you accept that between > two points their is always another? Why or why not? > If you do accept that, then what possible meaning > does "two consecutive points" have? Any two you pick, > trying to get that elusive consecutive pair, will > always have another between them. Therefore they are > not consecutive. > > Cheers, > Joe N
If I accept that between two points there is always another, then there is the aforementioned problem of division of a straight-line in half, which I cannot understand if I admit that between two points there's always another. Therefore, unless you show me how we can solve the aforementioned problem with the admission that between two points there is another, I cannot help thinking that two consecutive points exist.