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.

Actually, the number of points on a line segment is C, the continuum. It is not known (and indeed, it is undecidable) whether or not C equals aleph1.

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