humphreys wrote:At1, out of interest, do you have an actual degree in mathematics or are you just an amateur fan, so to speak?

Not meaning to detract from your argument if you don't have one, I'm just interested if you have credentials.

I do have a degree in math

and I consider myself to be an amateur fan.

humphreys wrote:at1with0 wrote:Between two points there are infinitely many points in between and that is what I think is causing Frosty to assert that the distance between two points is infinite.

If two points are in exactly the same position, they should not be considered two separate points, do you agree?

Yes.

In which case, there must be a finite non-zero distance between every two points, yes?

If we're talking about points in a Euclidean space such as a line, plane, or 3d space, then between any two points there is a finite non-zero distance. In the "pathological" case when the 2nd point is the 1st point, there no longer are two points and the distance between a point and itself is zero.

What's weird is that in such a context, there are infinitely many points on the line segment connecting two points.

Any non-zero number multiplied by infinity must equal infinity?

It depends on how small that non-zero number is and how big that infinity is. For example, if the non-zero number is an infinitesimal and the infinity is, say, the reciprocal of that infinitesimal, then the product of the two is an appreciable, finite number. However, in many situations, depending on the two inputs, a non-zero number multiplied by infinity is infinity.

In which case, if there are an infinite number of points, the total distance must also be infinite. The total distance cannot actually be infinite, therefore there cannot be an infinite number of points, as far as I can see. Again, we have to accept the minimum size plank length to get around such paradoxes.

The length of a point is zero so we have zero times infinity which is "indeterminate," meaning that it is not a specific appreciable number.