This is a

paradox mentioned in the

Greek work

*Mechanica*,
which some attribute to

Aristotle.
Picture a

wheel with two

concentric circles of different

diameters - a wheel within a wheel.

/-\ /-\
| | | |
| O_______O |
| | | |
\_/_____\_/

There is a 1:1

correspondence of

points on the large circle with points
on the small circle. The larger wheel and the smaller wheel will both
travel the same distance when it is rolled. This seems to imply that
the two

circumferences of the different sized circles are

equal,
which is clearly

impossible.

The fallacy of this lies in the assumption that a 1:1 correspondence
of points means the two curves must be of equal length. Actually
the number of points on a line segment of any length are all the same:
aleph_{1}

Or more simply:
`y = 0`

`y = x`

/
/
/
+---

There is a 1:1 relationship with the number of points on these two lines,
however, the two lines are not the same length.