And so the set of rationals and its complementary set of irrationals are in one way comparable(they are both dense in the number line)and in another not(therst set is countable, the second not).
Cantor’s Middle Third Set
We now have a clearer idea as to how the rational and the irrational numbers interlace to form the real number line. The rational numbers form a countable set, yet are densely packed into the number line. Cantor’s Middle Third Set is, by way of contrast,an uncountable subset of the unit interval that nevertheless is sparsely spread. It is the result of the following construction.