Here is a proof that demonstrates that the space occupied by the rational numbers is the same as the space occupied the integers (There’s a related demonstration that the space of irrational numbers is larger, but I don’t remember it).

Whatever it is that Cantor proved, the proof goes something like this:

` 1. Enumerate the positive integers along the x-axis 2. Enumerate the positive integers along the y-axis 3. Mark the point (0, 0). 4. Mark the point (1, 1). 5. Draw a horizontal vector one unit to the right. 6. Draw a diagonal (up and to the left) vector to the line at x=1, marking each intersection. 7. Draw a vertical vector one unit up. 8. Draw a diagonal (down and to the right) vector to the line at y=1, marking each intersection. 9. Go to step 5.`

This algorithm traverses the following points (in Quadrant I):

(0,0), (1,1), (2,1), (1,2), (1,3), (2,2), (3,1), (4,1), (3,2), (2,3), (1,4), ...

Treating each x-coordinate as the numerator, and y-coordinate as the denominator (and removing duplicates), this algorithm traverses the following rational numbers:

(0), (1), (2), (1/2), (1/3), (3), (4), (3/2), (2/3), (1/4), ...

By construction, this algorithm traverses every point in Quadrant I for which the ratio of x/y is defined.

By symmetry, this algorithm also traverses every point in Quadrant II for which the ratio of x/y is defined.

Every ratio is touched, and every intersection traversed contains a ratio.

Therefore the size of the rational numbers is the same as the size of the natural (positive and negative) “counting numbers”.