Galileo's paradox is a demonstration of one of the surprising properties of
infinite sets. The ideas were not new with him, but his name has come to be associated with them. In his final scientific work,
Two New Sciences,
Galileo Galilei made apparently contradictory statements about the
positive integers. First, some numbers are
squares, while others are not; therefore, all the numbers, including both squares and non-squares, must be more numerous than just the squares. And yet, for every square there is exactly one positive number that is its
square root, and for every number there is exactly one square; hence, there cannot be more of one than of the other. This is an early use, though not the first, of the idea of
one-to-one correspondence in the context of infinite sets.