Quote:
Originally Posted by Anthony
The degenerate case of a point is not aleph-null -- the number of points on a surface is the same cardinality as the set of real numbers.
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That's true, but I'm not sure it's relevant.
Here, for example, is the interval [0, 1]. Now we look selectively at only the points in that interval that correspond to rational numbers (it could be algebraic numbers, but they're no more numerous than rational numbers, so let's stay with the simpler case). As we go through the ordered list of rational numbers that correspond to proper fractions (x/y, where x < y and both x and y are positive), the length of the line segment occupied by each point seems to go to zero as the number of points goes to infinity.
Similarly, on a spherical surface, the area occupied by each rationally numbered point seems to go to zero. The fact that those points do not make up the entirety of the spherical surface may not make a difference.