Galileo's paradox says that on one side there are fewer square numbers (second powers) among natural numbers than all numbers because only some numbers are squares. On the other side, there are as many squares as numbers because every square has its root and every number has its square.
There are two ways to construct the set of all squares of natural numbers in ZF.
- According to the axiom of specification, we consider the set $A = \{n \in \mathbb N, (\exists m)(m \in \mathbb N) \wedge n = m^2\}$.
- According to the replacement schema, if we use a function $F(x) = x^2$ we get the set $B = \{n^2, n \in \mathbb N\}$.
It seems that $A$ is a proper subset of natural numbers $A \subset \mathbb N$.
$B$ is an image of natural numbers, $B = F(A)$ but is $B \subseteq \mathbb N$?
Finally, is $A = B$?