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:
aleph1
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.