It is a very treacherous business to look at constructing an infinite structure from finite units, or for that matter, a finite structure from infinitesimal units or units with a measure equal to 0.Often in mathematics it is best to take the opposite approach, for example, instead of asking yourself how many points there are between two other points, or how many points it would take to make up the line between two other points, just reflect that there is no point on that line that one cannot represent with a suitably chosen number.

You also can think out other attributes of the line and points on that line, such as that no point on the line is in any way not on the line; it does not overlap anything else.

If you removed the part of the line corresponding to that point, you would have split the line into two segments. The length of the line segments would not have changed, but they would now be separate, and they also would have changed some of their attributes; for one thing each of these segments had previously ended at that point exactly. Now suddenly neither of them has an ending point at that end.

We say that each segment is open at the end terminating at that point, whereas previously it had been "closed". we say that a line segment is "open" when it has no last point at that end, and "closed" when it does have an identifiable last point. In some matters such things don't matter; in other matters they may matter crucially. 

Anyway, I hope you can see that by observing that a line behaves in crucial ways like a sequence of points and nothing but, we avoid any difficulties of arguing about how we would go about constructing a line from a lot of points of zero length.

What You have to know, is that the "number" of points on a continuum

is much "bigger" than the number of natural numbers.

Look here:

http://en.wikipedia.org/wiki/Infinity

Scrool down to:

Cardinality of the continuum

Remember that "mathematics" is but a mindsport (as well as a mind****.)

Fortunately, a small segment of it approximates our perception of "reality."

Use when useful.

If you have problems with points then you are invery good company, even Euclid had problems and defined a point as "that which has no parts". I have always comforted myself with the view that a cube is three dimensional, a square two dimensional, a line one dimensional and so it follows that a point is zero dimensional. There is, of course, a major problem if a point has a size and that is it would limit the accuracy with which we could measure the length of a line.

There is a branch of mathematics, which I believe started out as a philosophical discussion, called pointless geometry. This starts off with using regions rather than points. Which brings me to the old joke that without geometry our existance would be pointless.

Thank you all for your wonderful answers, which did confirm that the basis of our mathematical understanding of the world is at best philosophical. 

This question is haunting me since I started working on the principles of fractal Geometry (where a point is one dimension).

what bothered me is the value of zero in a fractal dimension. if a point is one dimension and it has zero dimensions, then zero has a specific value! 

so whether it is geometry or math, I always end with the concept of a mathematical dimension (not axiom!) where zero has a specific value(s) but that is another story.