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.
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 quantities but not to infinite quantities.
During the nineteenth century Cantor found a framework in which this restriction is not necessary; it is possible to define comparisons amongst infinite sets in a meaningful way (by which definition the two sets, integers and squares, have "the same size"), and that by this definition some infinite sets are strictly larger than others.