There is a one-to-one correspondence between the set of squares and the set of square roots. Simply map each number in the set of square roots to its square in the set of squares. So it looks as though the sets must have the same number of members. After all:

(1) If there is a one-to-one correspondence between S1 and S2, then S1 has the same number of members as S2.

And yet it is deeply intuitive that there are more square roots than there are squares. Take away the squares from the set of square roots, and you still have some numbers left over. Take away the square roots from the set of squares, on the other hand, and you are left with nothing. So it seems that there must be fewer squares than square roots. This suggests the following:

(2) If S1 is a proper subset of S2, then S1 has fewer members than S2

These two highly intuitive principles cannot both be correct. This was once known as Galileo’s paradox. Virtually no one seems to find it paradoxical anymore. There is a widespread confidence that, however counterintuitive the resolution might seem, the matter has been settled: (1) is true, so (2) must be false. But why reject (2) rather than (1)? What we have here is an intuitional conflict. Has anything actually happened, since Galileo, to make us think that the intuition that backs (1) is more likely to be correct than the intuition that backs (2)? Not that I know of. All that has happened is that there has been a decision to use one-to-one correspondence as a measure of equinumerosity.

In fact, it seems that there is positive reason to prefer (2) to (1). After all, we have clear, concrete-case intuitions to the effect that there are more natural numbers than even numbers, more multiples of three that multiples of six, more square roots than squares. One might even be so bold as to call these “counterexamples” to (1), which is the label we typically reserve for concrete-case intuitions that tell against a general principle, even when that general principle is itself intuitive. So, if anything, we should prefer (2) to (1).

Perhaps the best conclusion to draw is that there is a genuine paradox here, and that more work needs to be done in resolving the intuitional conflict. But it is hard to see why (1) is supposed to be sacrosanct, and why we ought to conclude that there are as many squares as square roots, given that our only evidence for (1) is intuitive, and that we have at least as much intuitive evidence for thinking that (1) is false as we do for thinking that it is true.

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