Galileo's paradox

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

In his final scientific work, the Two New Sciences, Galileo made two apparently contradictory statements about the positive whole numbers. First, some numbers are perfect 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 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 a proof by one-to-one correspondence of infinite sets.)

Galileo concluded that the ideas of less, equal, and greater applied only to finite sets, and did not make sense when applied to infinite sets. In the nineteenth century Cantor, using the same methods, showed that while Galileo's result was correct as applied to the whole numbers and even the rational numbers, the general conclusion did not follow: some infinite sets are larger than others, in that they cannot be put into one-to-one correspondence.

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