Galileo noticed there seem to be as many perfect squares as there are whole numbers
Every whole number (1, 2, 3...) can be paired with exactly one perfect square (1, 4, 9...) by squaring it, and every perfect square pairs back to exactly one whole number by taking its root, a one-to-one pairing that seems to prove the two sets are the same size. But perfect squares are also clearly a small, sparse subset of all whole numbers, since most numbers aren't perfect squares at all. How can both be true?
Reveal the answer
Galileo concluded in 1638 that for infinite sets, the ordinary rules of 'bigger than,' 'smaller than' and 'equal to' simply don't apply the way they do for finite collections. It took until the 1870s for Georg Cantor to formalize the insight, showing that an infinite set can indeed be 'the same size' as one of its own subsets, now the defining property mathematicians use to characterize infinity itself.