Galileo's Paradox

Galileo's paradox is a demonstration of one of the surprising properties of infinite sets. 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.

Galileo concluded that the ideas of less, equal, and greater apply to finite sets, but not to infinite sets. In the nineteenth century, using the same methods, Cantor showed that this restriction is not necessary. It is possible to define comparisons amongst infinite sets in a meaningful way (by which definition the two sets he considers, integers and squares, have "the same size"), and that by this definition some infinite sets are strictly larger than others.