Why does Galileo’s pairing of counting numbers with even numbers seem surprising?

Galileo’s pairing of counting numbers with even numbers is surprising because it challenges our ordinary intuition regarding the size of sets. In finite sets, a proper subset—such as the set of even numbers within the set of all counting numbers—must necessarily contain fewer elements than the original set.

However, Galileo Galilei demonstrated that for infinite sets, it is possible to establish a one-to-one correspondence between the set of all counting numbers and the set of their doubles (the even numbers) with no elements left over. This result, now known as Galileo’s paradox, reveals that infinite sets can be equivalent to their own proper subsets, a property that contradicts the behavior of finite collections.12

Because of this, Galileo concluded that the standard concepts of "greater than," "less than," or "equal to" cannot be applied to infinite quantities in the same way they are applied to finite ones. This nonintuitive behavior highlights why ordinary numerical intuition is often misleading when dealing with infinite sets.12