KQ: Sorry, I've been trying to keep up, but you just lost me.
A point is on a line.
JG: Only if it corresponds to a rational number.
KQ: You admit sqrt(2) is a point on a line.
JG: No, I don't and never have admitted to this. You can place a point immediately after the line segment that denote sqrt(2), but this point does not correspond to any number.
KQ: Yet when you remove it you don't leave a hole.
JG: Yes, because it's not corresponding to any number, that is, it has not been reified.
KQ: If you take something out of a set of things, there's what I would call a 'hole' in the set.
JG: Except there is no such thing as a set of points. A set of numbers, yes. But not a set of points. In order to have a set of points, ALL the points must have names.
KQ: I think that's in fact the common usage. A 'hole' doesn't result in a set of objects if only a particular TYPE of object is removed. Commonly. To put it another way, a hole is a hole is a hole. Use another word or phrase than 'hole'? Like, 'rational hole'? I admit it's clumsy but in the interests of clarity?
JG: That's correct. But then you know what elements you are dealing with. They all have names. Points with no names do not belong to any set.
KQ: Again, sorry, but please refresh my memory. Why are we talking about holes on the line at all, and why is it necessary for your argument to distinguish between rational holes and non-rational holes?
JG: Well, I tried to clarify this concept because you seem adamant that a line contains points, but in actual fact, a line never contains any points, unless they can be reified.