What paradox compares positive integers with their square numbers?

Galileo’s paradox compares the positive integers with the numbers that are perfect squares.

In Two New Sciences, Galileo noticed that every positive integer has a corresponding square: 1 pairs with 1, 2 with 4, 3 with 9, and so on. Yet the squares form only part of the positive integers, because numbers such as 2, 3, 5, and 6 are not squares. For finite collections, a proper part normally contains fewer members than the whole, but this intuition breaks down for infinite collections.

Galileo therefore argued that the ordinary categories of “less,” “equal,” and “greater” do not apply straightforwardly to infinite quantities. Centuries later, Georg Cantor’s theory of sets clarified the issue. The positive integers and the perfect squares have the same cardinality because they can be paired one-to-one, even though the squares are a proper subset.

This is not the Fermi paradox, which asks why we have not observed extraterrestrial civilizations. Russell’s paradox concerns set membership, and Zeno’s paradoxes concern motion and infinity. Galileo’s paradox is specifically an early reflection on infinite sets and one-to-one correspondence.