Galilei's Paradox: The claim that there are exactly as many squares as natural numbers, even though the squares are a proper subset of the natural numbers.

The paradox was first published by the Italian scholar Galileo Galilei around the year 1590. Apparently, he was so shocked by its discovery that he decided to suspend any further scrutiny of the infinite. What he found was that since every square can be correlated with its even square root, and vice versa, we can construct two bijectively correlated infinite sets as follows:

Set A (natural numbers):    { 1, 2, 3,  4,  5,   6,   7,  ... }

Set B (squares): { 1, 4, 9, 16, 25, 36, 49, ... }

Although set B is a proper subset of A, both sets obviously have the same number of elements, since each element of set A can be correlated with exactly one element of set B and vice versa. Richard Dedekind used this property to define the very notion of an infinite set: A set is infinite if and only if there is a bijective function from A onto some proper subset of A (see also ►Countability).


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