If a point has no dimension and no area how can there be space?

Solution 1:

Because length, area, volume, etc. are not ways of counting points, but ways of counting sizes relative to a unit. The structures of a topological space or measure space assign the property of being a blob of space to collections of points, rather than assigning a makeup of points to them. When you have a finite set of points in what you want to be blobs of space, the counting measure is appropriate. But not when you get to infinite sets. Consider the map that sends the $n^{\text{th}}$ odd number to $n$ and the $n^{\text{th}}$ even number to $-n$, starting from the $0^{\text{th}}$ even number. Clearly the number of points is equal between the natural numbers and the integers. But you might find it more appropriate for the natural numbers to be half the size of the integers. Furthermore, any two intervals, by number of points, would be equal in length if you were measuring space as the number of points you can count in a region. The point is, if you're axiomatizing your notion of relative size, that's not the same as acquiring a notion of point content where every point contributes the same amount to the size.

Solution 2:

In order to calculate the "sum" of lengths/areas/volumes etc. for a collection of subsets, you need the collection of subsets to be countable. But if you take a curve or surface etc. and look at all the points on the set, it is uncountable. So you can't reason that the total length/area/volume etc. is $0 + 0 + \ldots = 0$. If anything, you have to consider that e.g. for length of $S$ along the line you get $dx + dx+ \ldots$, "summed" over all points in $S$, which is indeed zero for a countable point set $S$ but for all points on a line segment you get an integral (not countable sum) of $dx$, which gives the length of the line segment.

Solution 3:

The issue you raise here is similar/related to things like:

Archimedes axiom: http://en.wikipedia.org/wiki/Archimedean_axiom and Zeno's Paradoxes http://en.wikipedia.org/wiki/Zeno%27s_paradoxes