Galileo's Paradox

Theorem
The natural numbers $\N$ are in one-to-one correspondence with their squares:

That is, the mapping:
 * $\forall n \in \N: f: n \mapsto n^2$

is a bijection.

Hence the set of square numbers is equinumerous to the set of natural numbers.

That is, a set is equinumerous to one of its proper subsets.

Resolution
This is a veridical paradox.

$\N$ is an infinite set.

The defining property of an infinite set is that it possesses proper subsets with which it is equinumerous.

Also see

 * Infinite Set Equivalent to Proper Subset