How did Galileo Galilei pair counting numbers with their squares?

In 1638, Galileo Galilei demonstrated that the set of counting numbers could be placed into a one-to-one correspondence with the set of their squares. By pairing each counting number $n$ with its square $n^2$, he established a matching where every element in the first set corresponds to exactly one element in the second set, and vice versa.12

This pairing, represented as $\{(n,n^2)|n\in \mathbb{N}\}$, highlights a counterintuitive property of infinite sets: they can be equivalent to a proper subset of themselves. Because of this result, which is now known as Galileo’s paradox, he concluded that the standard concepts of "less than," "equal to," and "greater than" do not apply to infinite quantities.12