The Number Line under the Microscope

In Chapter 7 we saw how the real number line was a densely packed mix of the rational and the irrational. If the rational points were blue and the irrational red, what would we see? There would be a red dot between every pair of blue dots, and a blue dot separating every pair of reds, so we might expect the overall effect to be one of uniform purple. On the other hand, the blue points form only a countable set, which has measure zero, compared with the remaining red points, so the effect of the red would surely swamp that of the blue making the latter invisible. Neither interpretation stands up to scrutiny as there is no physical experiment that could approximate the limiting behavior of which we are speaking. We need to think of the line more in its own terms.