As an eight year old boy, I played with my classmates on the playground of the saint Francis school. When the bell sounded, we took up our positions in rows: the children of each class in the row of their class, arranged in order of age; thus, six rows altogether.
There are mathematicians (called platonists) that call the real numbers (the numbers on the
line of numbers) their children, and dare to place all of them on an imaginary playground.
Two children are classmates, then, if and only if their difference x - y is a ratio p/q for
integral numbers p and q. Thus, pi and pi + 3/4 are classmates. Furthermore they choose from each
class an eldest pupil between 0 and 1. So pi - 3 could be the eldest pupil of the class of pi,
and sqrt(2) - 1/2 could be the eldest pupil of the class of sqrt(2).
Now you have to know that we can assign to each rational number p/q a unique natural number, in such a way that the assigned numbers are exactly all natural numbers 0,1,2,3,... (to the rational number 0 we assign the natural number 0). Therefore, we can place the children of each class in one (infinitely long) row indeed: the eldest pupil in front, 17 places behind the eldest pupil the number that differs from the eldest by the rational number to which the natural number 17 has been assigned, etc.
We could place the class of sqrt(2) in a row farthest to the left on the playground, with its eldest pupil sqrt(2) - 1/2 in front, and next to it the class of pi, with pi - 3 in front: we give the class of sqrt(2) natural classnumber 0 and the class of pi natural classnumber 1. But there exist so many real numbers that we cannot give to each class a unique natural classnumber. You need to be very brave indeed, then, if you want to place all these classes from the left side to the right on a playground, if only in your thoughts.
Let us give the platonists the benefit of the doubt, and think together with them for a while. All classes are standing in rows on the playground. Now they collect the eldest pupils of the classes (each with assigned number 0 and standing in front of its row) in a set V0, the pupils with assigned number 1 following immediately after the eldest pupils in a set V1, the pupils with assigned number 2 in a set V2, etc.
But these sets V0, V1, V2, ... are subsets of the set of real numbers on the line of numbers as well (imagine that they can go from the playground to their places on the line of numbers and vice versa). Subsets of the line of numbers often have a length: the interval [2,13], for example, has length 11. And the union of the intervals [1/2,1], [1/4,1/3], [1/6,1/5], ... (etc) has length 1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + 1/7 - .... = 0.693... The sets V0, V1, V2, .. all have equal length, for one finds V35 (for instance) by translating V0 along the line of numbers over a distance equal to the absolute value of the rational number to which the natural number 35 has been assigned. Which length have they?
Have they length 0? Then their union has length 0 + 0 + 0 + ... = 0, but this union is the whole line of numbers, which has infinite length; so this is impossible.
So, have they positive length? But if, for example, the rational number to which the natural number 8 has been assigned is between 0 and 1, then all numbers of V8 are between 0 and 2 (for the numbers of V0, the eldest pupils, all are between 0 and 1 as well). And there exists an infinite number of rational numbers between 0 and 1, so there exists an infinite number of sets Vi whose pupils all are real numbers between 0 and 2. If such a set has positive length, then the union of these sets has infinite length (they don't overlap), and yet this union is a subset of [0,2], so its length can be at most 2. So this is impossible too.
In short, we can not at all measure how long the set V0 of eldest pupils on the line of
numbers is: it HAS no length. Furthermore, the eldest pupils for which we can point to their
places on the line form in reality a negligible part of V0.
Does V0 properly exist ? It exists in the thoughts of these platonic mathematicians, who call themselves brave. Okay, Santa Claus exists as well, but he is not proof against severe investigation. We prefer to call those platonists reckless.