Thinking about infinity

[RoadRunner79]

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I have been thinking about this saying that "a line is made up of an infinite number of points" lately.  The point doesn't seem to be very well defined.   It seems that to me, that the only way you could get an infinite number of points, is if you have your point defined with zero size or length.  In which case, if you add them, you are not going to get anywhere. 
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Well, that's the thing.  A point is defined as having zero size and length.  When we talk about lines and points, we're not talking about drawing them (when you draw a line with a pencil, it's actually a really thin rectangle, and when you draw a point it's actually a really tiny circle); we're rather talking about the "ideal point", which we can approximate.  This is sort of related to what I said earlier about "arbitrary largeness"; the smaller your "tiny circle" gets when you draw a pencil (that is, the closer it gets to an ideal "point"), the more of them you'll be able to find on your line-that's-actually-a-thin-rectangle.

Ok... so we have an abstract non-real  point of zero size.  But what differentiates these points from one another?   How does that difference correlate to the length of the line?

Theoretically, you could place an infinity of these points on a single point (since they have no size).   However, if there is nothing to differentiate these points, then wouldn't it be more correct to say that there is one point rather than an infinity?