On Tuesday, February 18, 2014 10:57:51 AM UTC-5, John Gabriel wrote: > Just to address this statement in a bit more detail. > > > > KQ: You do grant that a point which is the marker for a rational distance can be removed and leave a hole in the line, but a point which is the marker for an irrational distance does not leave a hole if it is removed because it does not represent a rational distance (leaving aside for the moment the more philosophical question of how can you remove something and NOT leave a hole, if you have already granted its existence). > > > > JG: Yes, it's true. The part I do not agree with is: > > > > "leaving aside for the moment the more philosophical question of how can you remove something and NOT leave a hole, if you have already granted its existence" > > > > There is no contradiction here because you can add or remove points as you please, there are no holes. A hole will appear on a number line, if you remove a point corresponding to a rational number.
Sorry, I've been trying to keep up, but you just lost me.
A point is on a line. You admit sqrt(2) is a point on a line. Yet when you remove it you don't leave a hole. If you take something out of a set of things, there's what I would call a 'hole' in the set. 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?
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?