(Cross-posted at M-Phi)
In his Two New Sciences (1638), Galileo presents a puzzle about infinite collections of numbers that became known as ‘Galileo’s paradox’. Written in the form of a dialogue, the interlocutors in the text observe that there are many more positive integers than there are perfect squares, but that every positive integer is the root of a given square. And so, there is a one-to-one correspondence between the positive integers and the perfect squares, and thus we may conclude that there are as many positive integers as there are perfect squares. And yet, the initial assumption was that there are more positive integers than perfect squares, as every perfect square is a positive integer but not vice-versa; in other words, the collection of the perfect squares is strictly contained in the collection of the positive integers. How can they be of the same size then?
Galileo’s conclusion is that principles and concepts pertaining to the size of finite collections cannot be simply transposed, mutatis mutandis, to cases of infinity: “the attributes "equal," "greater," and "less," are not applicable to infinite, but only to finite, quantities.” With respect to finite collections, two uncontroversial principles hold:
Part-whole: a collection A that is strictly contained in a collection B has a strictly smaller size than B.
One-to-one: two collections for which there exists a one-to-one correspondence between their elements are of the same size.
What Galileo’s paradox shows is that, when moving to infinite cases, these two principles clash with each other, and thus that at least one of them has to go. In other words, we simply cannot transpose these two basic intuitions pertaining to counting finite collections to the case of infinite collections. As is well known, Cantor chose to keep One-to-one at the expenses of Part-whole, famously concluding that all countable infinite collections are of the same size (in his terms, have the same cardinality); this is still the reigning orthodoxy.