Are there as many even numbers as whole numbers?

Yes, by the definition of "as many" that mathematicians use for infinite collections: the two can be paired off one-to-one with nothing left over. Pair 1 with 2, 2 with 4, 3 with 6, and so on. Every whole number gets exactly one even partner, and every even number is the partner of exactly one whole number. Neither side runs out.

That sounds absurd, because the even numbers are only some of the whole numbers: half the numbers seem to be missing. Galileo met the same puzzle with squares in his final scientific work, Two New Sciences. Some numbers are squares and others are not, so all the numbers together must be more numerous than the squares alone. And yet every number has exactly one square, so there cannot be more of one than of the other. Both statements sound right, and they cannot both be true.

Galileo's way out, put in the mouth of his character Salviati, was to say that less, equal and greater apply to finite quantities but not to infinite ones. He was not the first to notice: Duns Scotus, about 1302, had compared the even numbers to the whole of the numbers.

In the nineteenth century Cantor took the other road. He defined comparisons between infinite sets by matching, so that the integers and the squares have the same size, and the same definition even shows that some infinite sets are strictly larger than others. In that framework the paradox is no longer a contradiction: for infinite collections, being a proper part of something no longer means being smaller than it.