How did Galileo's paradox compare counting numbers with their squares?

In the early 1600s, the Italian scientist Galileo Galilei addressed the nonintuitive nature of infinite sets through what is now known as Galileo’s paradox.

Galileo 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 ($1,2,3,\dots$) with its square ($1,4,9,\dots$), he showed that every number in the first set has a unique partner in the second, despite the fact that the squares appear to be a much smaller subset of the counting numbers. He similarly illustrated this by pairing counting numbers with their doubles (the set of even numbers).

Because of these results, which contradicted the intuition that a whole must be larger than its parts, Galileo concluded that it is impossible to speak of infinite quantities as being greater than, less than, or equal to one another. This paradox highlighted the limitations of applying finite logic to the concept of infinity.1