Take an interval of unit length and decompose it into a finite or even countable set of subintervals. Now rearrange the subintervals into an interval. What do we get? An interval of unit length.

Now think of our unit interval as made of uncountably many points. The first diagram shows how these points can be rearranged into an interval of length 2, or indeed a interval of any finite length.

We project the points in the unit interval standing on x=1 onto the interval of length 2 standing on x=2 using rays through the origin (resp. interval of length y standing on x=y).

The second diagram shows, using a slightly different projection, how we can rearrange a finite interval into a semi-infinite interval. The interval standing at 45 degrees becomes the whole of the positive x axis.

Finally, with a rather different mapping we can rearrange a unit interval into a set with no intervals at all. But as this is pretty hard to draw I refer you to Wikipedia's article on the Cantor set.

What's all this in aid of I hear you ask? Well, the idea is to persuade you that we can think of an interval as decomposable into finitely many, or even countably infinitely many, subintervals which behave pretty much as familiar mechanical parts. They can be taken apart and reassembled in a different order and we end up back with the same interval. There is, in other words, a well-behaved notion of 'adding together' that applies to subintervals. Contrast this with what happens when we think of our interval as made of uncountably many dimensionless points. We can take the points apart and rearrange them into intervals of any finite length we like, an infinite 'interval', or an 'interval' with two distinct endpoints but no length in between (that's the Cantor set). So there is no stable notion of 'adding together' that applies to points. For surely we want our idea of 'adding together' to give the same result whenever the same 'parts' are used?

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